{ "dataset": { "theorems": [ { "id": 0, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.1", "categories": [], "title": "The Triangle Inequality", "contents": [ "If $a$ and $b$ are any two real numbers$,$ then", "\\begin{equation} \\label{eq:1.1.3}", "|a+b|\\le |a|+|b|.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "There are four possibilities:", "\\begin{alist}", "\\item % (a)", "If $a\\ge0$ and $b\\ge0$, then $a+b\\ge0$, so", "$|a+b|=a+b=|a|+|b|$.", "\\item % (b)", "If $a\\le0$ and $b\\le0$, then $a+b\\le0$, so", "$|a+b|=-a+(-b)=|a|+|b|$.", "\\item % (c)", " If $a \\ge 0$ and $b \\le 0$, then $a+b=|a|-|b|$.", "\\item % (d)", " If $a \\le 0$ and $b \\ge 0$, then $a+b=-|a|+|b|$.", "\\end{alist}", "Eq.~\\ref{eq:1.1.3}", "holds in cases {\\bf (c)} and {\\bf (d)}, since", "\\begin{equation}", "|a+b|=", "\\begin{cases}", "|a|-|b|& \\text{ if } |a| \\ge |b|,\\\\", "|b|-|a|& \\text{ if } |b| \\ge |a|.", "\\end{cases}", "\\tag*{" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 1, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.3", "categories": [], "title": "", "contents": [ "If a nonempty set $S$ of real numbers is bounded above$,$ then", "$\\sup S$ is the unique real number $\\beta$ such that", "\\begin{alist}", "\\item % (a)", " $x\\le\\beta$ for all $x$ in $S;$", "\\item % (b)", " if $\\epsilon>0$ $($no matter how small$)$$,$ there is an $x_0$ in", "$S$ such that", "$x_0>", "\\beta-\\epsilon.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "We first show that $\\beta=\\sup S$ has properties \\part{a} and", "\\part{b}. Since $\\beta$ is an upper bound of $S$, it must satisfy", "\\part{a}. Since any real number $a$ less than $\\beta$ can be written", "as $\\beta-\\epsilon$ with $\\epsilon=\\beta-a>0$, \\part{b} is just", "another way of saying that no number less than $\\beta$ is an upper", "bound of $S$. Hence, $\\beta=\\sup S$ satisfies \\part{a} and \\part{b}.", "Now we show that there cannot be more than one real number with", "properties \\part{a} and \\part{b}. Suppose that $\\beta_1<\\beta_2$ and", "$\\beta_2$ has property \\part{b}; thus, if $\\epsilon>0$, there is an", "$x_0$ in $S$ such that $x_0>\\beta_2-\\epsilon$. Then, by taking", "$\\epsilon=\\beta_2-\\beta_1$, we see that there is an $x_0$ in $S$ such", "that", "$$", "x_0>\\beta_2-(\\beta_2-\\beta_1)=\\beta_1,", "$$", "so $\\beta_1$ cannot have property \\part{a}. Therefore, there cannot", "be more than one real number that satisfies both \\part{a} and", "\\part{b}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 2, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.4", "categories": [], "title": "", "contents": [ "If $\\rho$ and $\\epsilon$ are positive$,$ then $n\\epsilon>\\rho$ for", "some integer $n.$" ], "refs": [], "proofs": [ { "contents": [ "The proof is by contradiction.", "If the statement is false, $\\rho$ is an upper bound of", "the set", "$$", "S=\\set{x}{x=n\\epsilon,\\mbox{$n$ is an integer}}.", "$$", "Therefore, $S$ has a supremum $\\beta$, by property \\part{I}.", "Therefore,", "\\begin{equation}\\label{eq:1.1.9}", "n\\epsilon\\le\\beta \\mbox{\\quad for all integers $n$}.", "\\end{equation}}", "\\newpage\\noindent", "Since $n+1$ is an integer whenever $n$ is, \\eqref{eq:1.1.9} implies that", "$$", "(n+1)\\epsilon\\le\\beta", "$$", " and therefore", "$$", "n\\epsilon\\le\\beta-\\epsilon", "$$", " for all integers $n$. Hence,", " $\\beta-\\epsilon$ is an upper bound of $S$. Since $\\beta-\\epsilon", "<\\beta$, this contradicts the definition of~$\\beta$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.6", "categories": [], "title": "", "contents": [ "The rational numbers are dense in the reals$\\,;$ that is, if $a$", "and", "$b$ are real numbers with $a1$. There is also an integer", "$j$ such that $j>qa$. This is obvious if $a\\le0$, and it follows from", "Theorem~\\ref{thmtype:1.1.4} with $\\epsilon=1$ and $\\rho=qa$ if $a>0$. Let", "$p$ be the smallest integer such that $p>qa$. Then $p-1\\le qa$, so", "$$", "qa0$ $($no matter how small$\\,)$, there is an $x_0$ in $S$", "such that", "$x_0<", "\\alpha+\\epsilon.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "(Exercise~\\ref{exer:1.1.6})", "A set $S$ is {\\it bounded\\/} if", "there are numbers", "$a$ and", "$b$ such", "that $a\\le x\\le b$ for all $x$ in $S$. A bounded nonempty set has a", "unique supremum and a unique infimum, and", "\\begin{equation}\\label{eq:1.1.11}", "\\inf S\\le\\sup S", "\\end{equation}", "(Exercise~\\ref{exer:1.1.7})." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 6, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.2.1", "categories": [], "title": "Principle of Mathematical Induction", "contents": [ " Let $P_1,$ $P_2, $\\dots$,$ $P_n,$ \\dots\\ be", "propositions$,$ one", "for each positive integer$,$ such that", "\\begin{alist}", "\\item % (a)", " $P_1$ is true$;$", "\\item % (b)", " for each positive integer $n,$ $P_n$ implies $P_{n+1}.$", "\\end{alist}", "Then $P_n$ is true for each positive integer $n.$" ], "refs": [], "proofs": [ { "contents": [ "Let", "$$", "\\mathbb M=\\set{n}{n\\in \\mathbb N\\mbox{ and } P_n\\mbox{ is", "true}}.", "$$", "From \\part{a}, $1\\in \\mathbb M$, and from \\part{b}, $n+1\\in \\mathbb M$ whenever", "$n\\in \\mathbb M$. Therefore, $\\mathbb M=\\mathbb N$, by postulate", "\\part{E}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 7, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.2.2", "categories": [], "title": "", "contents": [ " Let $n_0$ be any integer $($positive$,$", " negative$,$ or zero$)$$.$ Let", "$P_{n_0},$ $P_{n_0+1},$ \\dots$,$ $P_n,$ \\dots\\ be propositions$,$", " one for each integer $n\\ge n_0,$ such that", "\\begin{alist}", "\\item % (a)", " $P_{n_0}$ is true$\\,;$", "\\item % (b)", " for each integer $n\\ge n_0,$ $P_n$ implies $P_{n+1}.$", "\\end{alist}", "Then $P_n$ is true for every integer $n\\ge n_0.$" ], "refs": [], "proofs": [ { "contents": [ "For $m\\ge1$, let $Q_m$ be the proposition defined by", "$Q_m=P_{m+n_0-1}$. Then $Q_1=P_{n_0}$ is true by \\part{a}.", "If $m\\ge1$ and $Q_m=P_{m+n_0-1}$ is true, then $Q_{m+1}=P_{m+n_0}$", "is true by \\part{b} with $n$ replaced by $m+n_0-1$. Therefore,", "$Q_m$ is true for all $m\\ge1$ by Theorem~\\ref{thmtype:1.2.1} with $P$", "replaced by $Q$ and $n$ replaced by $m$. This is equivalent", "to the statement that $P_n$ is true for all $n\\ge n_0$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:1.2.1" ], "ref_ids": [ 6 ] } ], "ref_ids": [] }, { "id": 8, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.2.3", "categories": [], "title": "", "contents": [ " Let $n_0$ be any integer $($positive$,$", " negative$,$ or zero$)$$.$ Let", "$P_{n_0},$ $P_{n_0+1}, $\\dots$,$ $P_n,$ \\dots\\ be propositions$,$", " one for each integer $n\\ge n_0,$ such that", "\\begin{alist}", "\\item % (a)", " $P_{n_0}$ is true$\\,;$", "\\item % (b)", "for $n\\ge n_0,$ $P_{n+1}$ is true if $P_{n_0},$ $P_{n_0+1}, $\\dots$,$", "$P_n$ are all true.", "\\end{alist}", "Then $P_n$ is true for $n\\ge n_0.$" ], "refs": [], "proofs": [ { "contents": [ "For $n\\ge n_0$, let $Q_n$ be the proposition that", " $P_{n_0}$, $P_{n_0+1}$, \\dots, $P_n$ are all true.", "Then $Q_{n_0}$ is true by \\part{a}. Since $Q_n$ implies $P_{n+1}$", "by \\part{b}, and $Q_{n+1}$ is true if $Q_n$ and $P_{n+1}$ are both true,", "Theorem~\\ref{thmtype:1.2.2} implies that $Q_n$ is true for all $n\\ge", "n_0$. Therefore, $P_n$ is true for all $n\\ge n_0$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:1.2.2" ], "ref_ids": [ 7 ] } ], "ref_ids": [] }, { "id": 9, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.3", "categories": [], "title": "", "contents": [ "\\begin{alist}", "\\item % (a)", " The union of open sets is open$.$", "\\item % (b)", " The intersection of closed sets is closed$.$", "\\end{alist}", "These statements apply to", "arbitrary collections, finite or infinite, of open and closed", "sets$.$" ], "refs": [], "proofs": [ { "contents": [ "\\part{a} Let ${\\mathcal G}$ be a collection of open sets and", "$$", "S=\\cup\\set{G}{G\\in {\\mathcal G}}.", "$$", "If $x_0\\in S$, then $x_0\\in G_0$ for some $G_0$ in ${\\mathcal G}$, and", "since $G_0$ is open, it contains some $\\epsilon$-neighborhood of", "$x_0$. Since $G_0\\subset S$, this $\\epsilon$-neighborhood is in $S$,", "which is consequently a neighborhood of $x_0$. Thus, $S$ is a", "neighborhood of each of its points, and therefore open, by definition.", "\\part{b} Let ${\\mathcal F}$ be a collection of closed sets and $T", "=\\cap\\set{F}{F\\in {\\mathcal F}}$. Then $T^c=\\cup\\set{F^c}{F\\in {\\mathcal", "F}}$", "(Exercise~\\ref{exer:1.3.7}) and, since each $F^c$ is open,", "$T^c$ is open, from \\part{a}. Therefore, $T$ is closed, by", "definition." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 10, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.5", "categories": [], "title": "", "contents": [ "no point of $S^c$ is a limit point of~$S.$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $S$ is closed and $x_0\\in S^c$. Since $S^c$ is open,", "there is a neighborhood of $x_0$ that is contained in $S^c$ and", "therefore contains no points of $S$. Hence, $x_0$ cannot be a limit", "point of $S$. For the converse, if no point of $S^c$ is a limit point", "of $S$ then every point in $S^c$ must have a neighborhood contained", "in $S^c$. Therefore, $S^c$ is open and $S$ is closed." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 11, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.7", "categories": [], "title": "", "contents": [ "If ${\\mathcal H}$ is an open covering of a closed and bounded subset $S$", "of the real line$,$ then $S$ has an open covering $\\widetilde{\\mathcal", "H}$ consisting of finitely many open sets belonging to ${\\mathcal H}.$" ], "refs": [], "proofs": [ { "contents": [ "Since $S$ is bounded, it has an infimum $\\alpha$", "and a supremum $\\beta$, and, since $S$ is closed, $\\alpha$", "and $\\beta$ belong to $S$ (Exercise~\\ref{exer:1.3.17}). Define", "$$", "S_t=S\\cap [\\alpha,t] \\mbox{\\quad for \\ } t\\ge\\alpha,", "$$", "and let", "$$", "F=\\set{t}{\\alpha\\le t\\le\\beta \\mbox{\\ and finitely many sets from", "${\\mathcal H}$ cover $S_t$}}.", "$$", "Since $S_\\beta=S$, the theorem will be proved if we can show that", "$\\beta", "\\in F$. To do this, we use the completeness of the reals.", "Since $\\alpha\\in S$, $S_\\alpha$ is the singleton set $\\{\\alpha\\}$,", "which is contained in some open set $H_\\alpha$ from ${\\mathcal H}$", "because ${\\mathcal H}$ covers $S$; therefore, $\\alpha\\in F$. Since $F$ is", "nonempty and bounded above by $\\beta$, it has a supremum $\\gamma$.", "First, we wish to show that $\\gamma=\\beta$. Since $\\gamma\\le\\beta$ by", "definition of $F$, it suffices to rule out the possibility that", "$\\gamma<\\beta$. We consider two cases.", "{\\sc Case 1}. Suppose that $\\gamma<\\beta$ and $\\gamma\\not\\in S$. Then,", "since $S$ is closed, $\\gamma$ is not a limit point of $S$", "(Theorem~\\ref{thmtype:1.3.5}). Consequently, there is an $\\epsilon>0$", "such that", "$$", "[\\gamma-\\epsilon,\\gamma+\\epsilon]\\cap S=\\emptyset,", "$$", "so $S_{\\gamma-\\epsilon}=S_{\\gamma+\\epsilon}$. However, the", "definition of $\\gamma$ implies that $S_{\\gamma-\\epsilon}$ has a finite", "subcovering from ${\\mathcal H}$, while $S_{\\gamma+\\epsilon}$ does not.", "This is a contradiction.", "{\\sc Case 2}. Suppose that $\\gamma<\\beta$ and $\\gamma\\in S$. Then", "there is an open", "set $H_\\gamma$ in ${\\mathcal H}$ that contains $\\gamma$ and, along with $\\gamma$, an", "interval $[\\gamma-\\epsilon,\\gamma+\\epsilon]$ for some positive", "$\\epsilon$.", "Since $S_{\\gamma-\\epsilon}$ has a finite covering $\\{H_1, \\dots,H_n\\}$ of", "sets from ${\\mathcal H}$, it follows that $S_{\\gamma+\\epsilon}$ has the finite", "covering $\\{H_1, \\dots,H_n,H_\\gamma\\}$. This contradicts the", "definition of $\\gamma$.", "Now we know that $\\gamma=\\beta$, which is in $S$. Therefore, there is", "an open set $H_\\beta$ in ${\\mathcal H}$ that contains $\\beta$ and along", "with $\\beta$, an interval of the form", "$[\\beta-\\epsilon,\\beta+\\epsilon]$, for some positive $\\epsilon$. Since", "$S_{\\beta-\\epsilon}$ is covered by a finite collection of sets", "$\\{H_1, \\dots,H_k\\}$, $S_\\beta$ is covered by the finite collection", "$\\{H_1, \\dots, H_k, H_\\beta\\}$. Since $S_\\beta=S$, we are", "finished." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:1.3.5" ], "ref_ids": [ 10 ] } ], "ref_ids": [] }, { "id": 12, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.8", "categories": [], "title": "", "contents": [ " Every bounded infinite set of real numbers has at least one", "limit point$.$" ], "refs": [], "proofs": [ { "contents": [ "We will show that a bounded nonempty set without a limit point", "can contain only a finite number of points. If $S$ has no limit", "points, then $S$ is closed (Theorem~\\ref{thmtype:1.3.5}) and every point", "$x$ of $S$ has an open neighborhood $N_x$ that contains no point of", "$S$ other than $x$. The collection", "$$", "{\\mathcal H}=\\set{N_x}{x\\in S}", "$$", "is an open covering for $S$. Since $S$ is also bounded,", "Theorem~\\ref{thmtype:1.3.7} implies that $S$ can be covered by a finite", "collection of sets from ${\\mathcal H}$, say $N_{x_1}$, \\dots, $N_{x_n}$.", "Since", "these sets contain only $x_1$, \\dots, $x_n$ from $S$, it follows that", "$S=\\{x_1, \\dots,x_n\\}$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:1.3.5", "TRENCH_REAL_ANALYSIS-thmtype:1.3.7" ], "ref_ids": [ 10, 11 ] } ], "ref_ids": [] }, { "id": 13, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.3", "categories": [], "title": "", "contents": [ "then it is unique$\\,;$ that is$,$ if", "\\begin{equation} \\label{eq:2.1.7}", "\\lim_{x\\to x_0} f(x)=L_1\\mbox{\\quad and \\quad}\\lim_{x\\to x_0} f(x)=", "L_2,", "\\end{equation}", "then $L_1=L_2.$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that \\eqref{eq:2.1.7} holds and let $\\epsilon>0$.", "From Definition~\\ref{thmtype:2.1.2}, there are", "positive numbers $\\delta_1$ and $\\delta_2$ such that", "$$", "|f(x)-L_i|<\\epsilon\\mbox{\\quad if \\quad} 0<|x-x_0|<\\delta_i,", "\\quad i=1,2.", "$$", "If $\\delta=\\min(\\delta_1,\\delta_2)$, then", "\\begin{eqnarray*}", "|L_1-L_2|\\ar= |L_1-f(x)+f(x)-L_2|\\\\", "\\ar \\le|L_1-f(x)|+|f(x)-L_2|<2\\epsilon", "\\mbox{\\quad if \\quad} 0<|x-x_0|<\\delta.", "\\end{eqnarray*}", "We have now established an inequality that does not depend on $x$;", "that is,", "$$", "|L_1-L_2|<2\\epsilon.", "$$", "Since this holds for any positive $\\epsilon$,", " $L_1=L_2$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.1.2" ], "ref_ids": [ 303 ] } ], "ref_ids": [] }, { "id": 14, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.4", "categories": [], "title": "", "contents": [ "\\begin{equation}\\label{eq:2.1.9}", "\\lim_{x\\to x_0} f(x)=L_1\\mbox{\\quad and \\quad}\\lim_{x\\to x_0} g(x)=", "L_2,", "\\end{equation}", "then", "\\begin{eqnarray}", "\\lim_{x\\to x_0} (f+g)(x)\\ar= L_1+L_2,\\label{eq:2.1.10}\\\\", "\\lim_{x\\to x_0} (f-g)(x)\\ar= L_1-L_2,\\label{eq:2.1.11}\\\\", "\\lim_{x\\to x_0} (fg)(x)\\ar= L_1L_2,\\label{eq:2.1.12}\\\\", "\\arraytext{and, if $L_2\\ne0$,}\\\\", "\\lim_{x\\to x_0}\\left(\\frac{f}{g}\\right)(x)\\ar= \\frac{L_1}{", "L_2}.\\label{eq:2.1.13}", "\\end{eqnarray}" ], "refs": [], "proofs": [ { "contents": [ "From \\eqref{eq:2.1.9} and Definition~\\ref{thmtype:2.1.2},", " if $\\epsilon>0$, there is a", "$\\delta_1>0$ such that", "\\begin{equation}\\label{eq:2.1.14}", "|f(x)-L_1|<\\epsilon", "\\end{equation}", "if $0<|x-x_0|<\\delta_1$, and a $\\delta_2>0$ such that", "\\begin{equation}\\label{eq:2.1.15}", "|g(x)-L_2|<\\epsilon", "\\end{equation}", "if $0<|x-x_0|<\\delta_2$. Suppose that", "\\begin{equation}\\label{eq:2.1.16}", "0<|x-x_0|<\\delta=\\min (\\delta_1,\\delta_2),", "\\end{equation}", "so that \\eqref{eq:2.1.14} and \\eqref{eq:2.1.15} both hold. Then", "\\begin{eqnarray*}", "|(f\\pm g)(x)-(L_1\\pm L_2)|\\ar= |(f(x)-L_1)\\pm", "(g(x)-L_2)|\\\\", "\\ar \\le|f(x)-L_1|+|g(x)-L_2|<2\\epsilon,", "\\end{eqnarray*}", "which proves \\eqref{eq:2.1.10} and \\eqref{eq:2.1.11}.", "To prove \\eqref{eq:2.1.12}, we assume \\eqref{eq:2.1.16} and write", "\\begin{eqnarray*}", "|(fg)(x)-L_1L_2|\\ar= |f(x)g(x)-L_1L_2|\\\\[.5\\jot]", "\\ar= |f(x)(g(x)-L_2)+L_2(f(x)-L_1)|\\\\[.5\\jot]", "\\ar \\le|f(x)||g(x)-L_2|+|L_2||f(x)-L_1|\\\\[.5\\jot]", "\\ar \\le(|f(x)|+|L_2|)\\epsilon\\mbox{\\quad (from \\eqref{eq:2.1.14} and", "\\eqref{eq:2.1.15})}\\\\[.5\\jot]", "\\ar \\le(|f(x)-L_1|+|L_1|+|L_2|)\\epsilon\\\\[.5\\jot]", "\\ar \\le(\\epsilon+|L_1|+|L_2|)\\epsilon\\mbox{\\quad from", "\\eqref{eq:2.1.14}}\\\\[.5\\jot]", "\\ar \\le (1+|L_1|+|L_2|)\\epsilon", "\\end{eqnarray*}", "if $\\epsilon<1$", "and $x$ satisfies \\eqref{eq:2.1.16}. This proves", "\\eqref{eq:2.1.12}.", "To prove \\eqref{eq:2.1.13}, we first observe that if $L_2\\ne0$, there is", "a $\\delta_3>0$ such that", "$$", "|g(x)-L_2|<\\frac{|L_2|}{2},", "$$", "so", "\\begin{equation} \\label{eq:2.1.17}", "|g(x)|>\\frac{|L_2|}{2}", "\\end{equation}", "if", "$$", "0<|x-x_0|<\\delta_3.", "$$", "To see this, let $L=L_2$ and $\\epsilon=|L_2|/2$ in", "\\eqref{eq:2.1.4}. Now suppose that", "$$", "0<|x-x_0|<\\min", "(\\delta_1,\\delta_2,\\delta_3),", "$$", "\\nopagebreak", " so that \\eqref{eq:2.1.14}, \\eqref{eq:2.1.15},", "and \\eqref{eq:2.1.17} all hold. Then", "\\pagebreak", "\\begin{eqnarray*}", "\\left|\\left(\\frac{f}{ g}\\right)(x)-\\frac{L_1}{ L_2}\\right|", "\\ar= \\left|\\frac{f(x)}{ g(x)}-\\frac{L_1}{ L_2}\\right|\\\\", "\\ar= \\frac{|L_2f(x)-L_1g(x)|}{|g(x)L_2|}\\\\", "\\ar \\le\\frac{2}{ |L_2|^2}|L_2f(x)-L_1g(x)|\\\\", "\\ar= \\frac{2}{ |L_2|^2}\\left|L_2[f(x)-L_1]+", "L_1[L_2-g(x)]\\right|\\mbox{\\quad (from \\eqref{eq:2.1.17})}\\\\", "\\ar \\le\\frac{2}{ |L_2|^2}\\left[|L_2||f(x)-L_1|+|L_1|", "|L_2-g(x)|\\right]\\\\", "\\ar \\le\\frac{2}{ |L_2|^2}(|L_2|+|L_1|)\\epsilon", "\\mbox{\\quad (from \\eqref{eq:2.1.14} and \\eqref{eq:2.1.15})}.", "\\end{eqnarray*}", "This proves \\eqref{eq:2.1.13}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.1.2" ], "ref_ids": [ 303 ] } ], "ref_ids": [] }, { "id": 15, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.6", "categories": [], "title": "", "contents": [ "A function $f$ has a limit at $x_0$", "if and only if it has left- and right-hand limits at $x_0,$ and they", "are equal. More specifically$,$", "$$", "\\lim_{x\\to x_0} f(x)=L", "$$", "if and only if", "$$", "f(x_0+)=f(x_0-)=L.", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 16, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.9", "categories": [], "title": "", "contents": [ "Suppose that $f$ is monotonic on $(a,b)$ and define", "$$", "\\alpha=\\inf_{a\\alpha$, there is an $x_0$ in $(a,b)$ such that $f(x_0)-\\infty$, let", "$M=\\alpha+\\epsilon$, where $\\epsilon>0$. Then $\\alpha\\le", "f(x)<\\alpha+\\epsilon$, so", "\\begin{equation} \\label{eq:2.1.20}", "|f(x)-\\alpha|<\\epsilon\\mbox{\\quad if \\quad} a-\\infty$, let $\\delta=x_0-a$. Then \\eqref{eq:2.1.20} is equivalent to", "$$", "|f(x)-\\alpha|<\\epsilon\\mbox{\\quad if \\quad} aM$. Since $f$ is nondecreasing, $f(x)>M$ if", "$x_00$. Then", "$\\beta-\\epsilon< f(x)\\le\\beta$, so", "\\begin{equation} \\label{eq:2.1.21}", "|f(x)-\\beta|<\\epsilon\\mbox{\\quad if \\quad} x_00$, there is an $a_1$ in $[a,x_0)$ such that", "\\begin{equation} \\label{eq:2.1.22}", "f(x)<\\beta+\\epsilon\\mbox{\\quad if \\quad} a_1\\le x0$ and $a_1$ is in $[a,x_0),$ then", "$$", "f(\\overline x)>\\beta-\\epsilon\\mbox{\\quad for some }\\overline", "x\\in[a_1,x_0).", "$$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is bounded on $[a,x_0)$, $S_f(x;x_0)$ is nonincreasing and", "bounded on $[a,x_0)$. By applying Theorem~\\ref{thmtype:2.1.9}\\part{b} to", "$S_f(x;x_0)$, we conclude that $\\beta$", "exists (finite). Therefore, if $\\epsilon>0$, there is an $\\overline a$", "in", "$[a,x_0)$ such that}", "\\begin{equation} \\label{eq:2.1.23}", "\\beta-\\epsilon/2\\beta-\\epsilon/2.", "\\end{equation}", "Since $S_f(x_1;x_0)$ is the supremum of $\\set{f(t)}{x_1S_f(x_1;x_0)-\\epsilon/2.", "$$", "This and \\eqref{eq:2.1.24} imply that $f(\\overline x)>\\beta-\\epsilon$.", "Since $\\overline x$ is in $[a_1,x_0)$, this proves \\part{b}.", "Now we show that there cannot be more than one real number with", "properties \\part{a} and \\part{b}. Suppose that $\\beta_1<\\beta_2$ and", "$\\beta_2$ has property \\part{b}; thus, if $\\epsilon>0$ and $a_1$ is", "in $[a,x_0)$, there is an", "$\\overline x$ in $[a_1,x_0)$ such that", "$f(\\overline x)>\\beta_2-\\epsilon$. Letting", "$\\epsilon=\\beta_2-\\beta_1$, we see that there is an $\\overline x$ in", " $[a_1,b)$ such that", "$$", "f(\\overline x)>\\beta_2-(\\beta_2-\\beta_1)=\\beta_1,", "$$", "so $\\beta_1$ cannot have property \\part{a}. Therefore, there cannot", "be more than one real number that satisfies both \\part{a} and", "\\part{b}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.1.9" ], "ref_ids": [ 16 ] } ], "ref_ids": [] }, { "id": 18, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.12", "categories": [], "title": "", "contents": [ "If $f$ is bounded on $[a,x_0),$", "then $\\alpha=\\liminf_{x\\to x_0-}f(x)$ exists", "and is the unique real number with the following properties:", "\\begin{alist}", "\\item % (a)", "If $\\epsilon>0,$ there is an $a_1$ in $[a,x_0)$ such that", "$$", "f(x)>\\alpha-\\epsilon\\mbox{\\quad if \\quad} a_1\\le x0$ and $a_1$ is in $[a,x_0),$ then", "$$", "f(\\overline x)<\\alpha+\\epsilon\\mbox{\\quad for some }\\overline", "x\\in[a_1,x_0).", "$$", "\\end{alist}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 19, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.2", "categories": [], "title": "", "contents": [ "\\vspace*{6pt}", "\\begin{alist}", "\\item % (a)", "A function $f$ is continuous at $x_0$ if and only if $f$ is defined on", "an open interval $(a,b)$ containing $x_0$ and for each", "$\\epsilon>0$ there is a $\\delta >0$ such that", "\\begin{equation}\\label{eq:2.2.1}", "|f(x)-f(x_0)|<\\epsilon", "\\end{equation}", "whenever $|x-x_0|<\\delta.$", "\\item % (b)", "A function $f$ is continuous from the right at $x_0$ if and only if", "$f$ is defined on an interval $[x_0,b)$ and for each $\\epsilon>", "0$", "there is a $\\delta>0$ such that $\\eqref{eq:2.2.1}$ holds whenever $x_0\\le", "x0$", "there is a $\\delta>0$ such that $\\eqref{eq:2.2.1}$ holds whenever", "$x_0-\\delta0$. Since $g(x_0)$ is an interior", "point of $D_f$ and $f$ is continuous at $g(x_0)$, there is a", "$\\delta_1>0$ such that $f(t)$ is defined and", "\\begin{equation}\\label{eq:2.2.4}", "|f(t)-f(g(x_0))|<\\epsilon\\mbox{\\quad if \\quad} |t-g(x_0)|<", "\\delta_1.", "\\end{equation}", "Since $g$ is continuous at $x_0$, there is a $\\delta>0$ such that", "$g(x)$ is defined and", "\\begin{equation}\\label{eq:2.2.5}", "|g(x)-g(x_0)|<\\delta_1\\mbox{\\quad if \\quad}|x-x_0|<\\delta.", "\\end{equation}", "Now \\eqref{eq:2.2.4} and \\eqref{eq:2.2.5} imply that", "$$", "|f(g(x))-f(g(x_0))|<\\epsilon\\mbox{\\quad if \\quad}|x-x_0|<\\delta.", "$$", " Therefore, $f\\circ g$ is continuous at $x_0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 22, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.8", "categories": [], "title": "", "contents": [ "If $f$ is continuous on a finite closed interval $[a,b],$ then $f$ is", "bounded on $[a,b].$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $t\\in [a,b]$. Since $f$ is continuous at $t$,", "there is an open interval $I_t$ containing $t$ such", "that", "\\begin{equation}\\label{eq:2.2.7}", "|f(x)-f(t)|<1 \\mbox{\\quad if \\quad}\\ x\\in I_t\\cap [a,b].", "\\end{equation}", "(To see this, set $\\epsilon=1$ in \\eqref{eq:2.2.1},", "Theorem~\\ref{thmtype:2.2.2}.) The collection", "${\\mathcal H}=\\set{I_t}{a\\le t\\le b}$", "is an open covering of $[a,b]$. Since $[a,b]$ is compact, the", "Heine--Borel theorem implies that there are finitely many points", "$t_1$,", "$t_2$, \\dots, $t_n$ such that the intervals $I_{t_1}$,", "$I_{t_2}$, \\dots, $I_{t_n}$", "cover $[a,b]$. According to \\eqref{eq:2.2.7} with $t=t_i$,", "$$", "|f(x)-f(t_i)|<1\\mbox{\\quad if \\quad}\\ x\\in I_{t_i}\\cap [a,b].", "$$", "Therefore,", "\\begin{equation}\\label{eq:2.2.8}", "\\begin{array}{rcl}", "|f(x)|\\ar =|(f(x)-f(t_i))+f(t_i)|\\le|f(x)-f(t_i)|+|f(t_i)|\\\\[2\\jot]", "\\ar\\le 1+|f(t_i)|\\mbox{\\quad if \\quad}\\", "x\\in I_{t_i}\\cap[a,b].", "\\end{array}", "\\end{equation}", " Let", "$$", "M=1+\\max_{1\\le i\\le n}|f(t_i)|.", "$$", "Since $[a,b]\\subset\\bigcup^n_{i=1}\\left(I_{t_i}\\cap", "[a,b]\\right)$, \\eqref{eq:2.2.8} implies that", "$|f(x)|\\le M$ if $x\\in [a,b]$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.2.2" ], "ref_ids": [ 19 ] } ], "ref_ids": [] }, { "id": 23, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.9", "categories": [], "title": "", "contents": [ "Suppose that $f$ is continuous on a finite closed interval $[a,b].$ Let", "$$", "\\alpha=\\inf_{a\\le x\\le b}f(x)\\mbox{\\quad and", "\\quad}\\beta=\\sup_{a\\le x\\le b}f(x).", "$$", "Then $\\alpha$ and $\\beta$ are respectively the minimum", "and maximum of $f$ on $[a,b];$ that is$,$", " there are points $x_1$ and $x_2$ in $[a,b]$ such that", "$$", "f(x_1)=\\alpha\\mbox{\\quad and \\quad} f(x_2)=\\beta.", "$$" ], "refs": [], "proofs": [ { "contents": [ "We show that $x_1$ exists and leave it to you to show that $x_2$", "exists (Exercise~\\ref{exer:2.2.24}).", "Suppose that there is no", "$x_1$ in $[a,b]$ such that $f(x_1)=\\alpha$. Then $f(x)>\\alpha$", "for all $x\\in[a,b]$. We will show that this leads to a", "contradiction.", "Suppose that $t\\in[a,b]$.", "Then $f(t)>\\alpha$, so", "$$", "f(t)>\\frac{f(t)+\\alpha}{2}>\\alpha.", "$$", "\\enlargethispage{1in}", "\\newpage", "\\noindent", "Since $f$ is continuous at $t$, there is an open interval $I_t$ about", "$t$ such that", "\\begin{equation}\\label{eq:2.2.9}", "f(x)>\\frac{f(t)+\\alpha}{2}\\mbox{\\quad if \\quad} x\\in", "I_t\\cap [a,b]", "\\end{equation}", "(Exercise~\\ref{exer:2.2.15}). The collection ${\\mathcal H}=\\set{I_t}{a\\le t\\le", "b}$ is an open covering of $[a,b]$. Since $[a,b]$ is compact, the", "Heine--Borel theorem implies that there are finitely many points $t_1$,", "$t_2$, \\dots, $t_n$ such that the intervals $I_{t_1}$,", "$I_{t_2}$, \\dots,", "$I_{t_n}$ cover $[a,b]$. Define", "$$", "\\alpha_1=\\min_{1\\le i\\le n}\\frac{f(t_i)+\\alpha}{2}.", "$$", "Then, since $[a,b]\\subset\\bigcup^n_{i=1} (I_{t_i}\\cap [a,b])$,", "\\eqref{eq:2.2.9} implies that", "$$", "f(t)>\\alpha_1,\\quad a\\le t\\le b.", "$$", "But $\\alpha_1>\\alpha$, so this contradicts the definition of $\\alpha$.", "Therefore, $f(x_1)=\\alpha$ for some $x_1$ in $[a,b]$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 24, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.10", "categories": [], "title": "Intermediate Value Theorem", "contents": [ "Suppose that $f$ is continuous on $[a,b],$ $f(a)\\ne f(b),$ and $\\mu$", "is between $f(a)$ and $f(b).$ Then $f(c)=\\mu$ for some", "$c$ in $(a,b).$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $f(a)<\\mu\\mu$, then $c>a$ and, since $f$ is", "continuous at $c$, there is an $\\epsilon>0$ such that", "$f(x)>\\mu$ if $c-\\epsilon0$ such that $f(x)<\\mu$ for $c\\le", "x0$. Since $f$ is continuous on $[a,b]$,", "for each $t$ in $[a,b]$ there is a positive number", "$\\delta_{t}$ such that", "\\begin{equation}\\label{eq:2.2.10}", "|f(x)-f(t)|<\\frac{\\epsilon}{2}", "\\mbox{\\quad if \\quad}", "|x-t|<2\\delta_{t}", "\\mbox{\\quad and \\quad} x\\in[a,b].", "\\end{equation}", "If $I_{t}=(t-\\delta_{t", "},t+\\delta_{t})$, the collection", "$$", "{\\mathcal H}=\\set{I_{t}}{t\\in [a,b]}", "$$", "is an open covering of $[a,b]$. Since $[a,b]$ is compact, the", "Heine--Borel theorem implies that there are finitely many points", "$t_1$, $t_2$, \\dots, $t_n$ in", "$[a,b]$ such that $I_{t_1}$, $I_{t_2}$, \\dots, $I_{t_n}$ cover", "$[a,b]$. Now define", "\\begin{equation}\\label{eq:2.2.11}", "\\delta=\\min\\{\\delta_{t_1},\\delta_{t_2}, \\dots,\\delta_{t_n}\\}.", "\\end{equation}", "We will show that if", "\\begin{equation} \\label{eq:2.2.12}", "|x-x'|<\\delta \\mbox{\\quad and \\quad}x,x'\\in [a,b],", "\\end{equation}", "then", "$|f(x)-f(x')|<\\epsilon$.", "From the triangle inequality,", "\\begin{equation} \\label{eq:2.2.13}", "\\begin{array}{rcl}", "|f(x)-f(x')|\\ar =", "|\\left(f(x)-f(t_r)\\right)+\\left(f(t_r)-f(x')\\right)|\\\\", "\\ar\\le |f(x)-f(t_r)|+|f(t_r)-f(x')|.", "\\end{array}", "\\end{equation}", "Since $I_{t_1}$, $I_{t_2}$, \\dots, $I_{t_n}$ cover $[a,b]$, $x$ must", "be in one of", "these intervals. Suppose that", "$x\\in I_{t_r}$; that is,", "\\begin{equation} \\label{eq:2.2.14}", "|x-t_r|<\\delta_{t_r}.", "\\end{equation}", "From \\eqref{eq:2.2.10} with $t=t_r$,", "\\begin{equation} \\label{eq:2.2.15}", "|f(x)-f(t_r)|<\\frac{\\epsilon}{2}.", "\\end{equation}", "From \\eqref{eq:2.2.12}, \\eqref{eq:2.2.14}, and the triangle inquality,", "$$", "|x'-t_r|=|(x'-x)+(x-t_r)|\\le", " |x'-x|+|x-t_r|<\\delta+\\delta_{t_r}\\le2\\delta_{t_r}.", "$$", "Therefore, \\eqref{eq:2.2.10} with $t=t_r$ and $x$ replaced by", "$x'$ implies that", "$$", "|f(x')-f(t_r)|<\\frac{\\epsilon}{2}.", "$$", "This, \\eqref{eq:2.2.13}, and \\eqref{eq:2.2.15} imply that", "$|f(x)-f(x')|<\\epsilon$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 26, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.14", "categories": [], "title": "", "contents": [ "If $f$ is monotonic and nonconstant on $[a,b],$ then $f$ is continuous", "on $[a,b]$ if and only if its range $R_f=\\set{f(x)}{x\\in[a,b]}$ is the", "closed interval with endpoints $f(a)$ and $f(b).$" ], "refs": [], "proofs": [ { "contents": [ "We assume that $f$ is nondecreasing, and", "leave the case where $f$ is nonincreasing to you", "(Exercise~\\ref{exer:2.2.34}).", "Theorem~\\ref{thmtype:2.1.9}\\part{a}", "implies that the set $\\widetilde R_f=\\set{f(x)}{x\\in(a,b)}$", "is a subset of the open interval $(f(a+),f(b-))$. Therefore,", "\\begin{equation} \\label{eq:2.2.16}", "R_f=\\{f(a)\\}\\cup\\widetilde", "R_f\\cup\\{f(b)\\}\\subset\\{f(a)\\}\\cup(f(a+),f(b-))\\cup\\{f(b)\\}.", "\\end{equation}", "Now", "suppose that $f$ is continuous on $[a,b]$. Then $f(a)=f(a+)$,", "$f(b-)=f(b)$,", "so \\eqref{eq:2.2.16} implies that", "$R_f\\subset[f(a),f(b)]$. If $f(a)<\\mu0$ such that", "$$", "|E(x)|<|f'(x_0)|\\mbox{\\quad if\\quad} |x-x_0|<\\delta,", "$$", "and the right side of \\eqref{eq:2.3.16} must have the same sign as", "$f'(x_0)$ for $|x-x_0|<\\delta$. Since the same is true of the left", "side, $f(x)-f(x_0)$ must change sign in every neighborhood of $x_0$", "(since $x-x_0$ does). Therefore, neither \\eqref{eq:2.3.14} nor", "\\eqref{eq:2.3.15} can hold for all $x$ in any interval about $x_0$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.3.2" ], "ref_ids": [ 244 ] } ], "ref_ids": [] }, { "id": 32, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.8", "categories": [], "title": "", "contents": [ "Suppose that $f$ is continuous on the closed interval $[a,b]$ and", "differentiable on the open interval $(a,b),$ and $f(a)=f(b).$ Then", "$f'(c)=0$ for some $c$ in the open interval $(a,b).$" ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is continuous on $[a,b]$, $f$ attains a maximum and a", "minimum", "value on $[a,b]$ (Theorem~\\ref{thmtype:2.2.9}). If these two", "extreme values are the same, then $f$ is constant on $(a,b)$, so", "$f'(x)=0$ for all $x$ in $(a,b)$. If the extreme values differ, then", "at least one must be attained at some point $c$ in the open interval", "$(a,b)$, and $f'(c)=0$, by Theorem~\\ref{thmtype:2.3.7}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.2.9", "TRENCH_REAL_ANALYSIS-thmtype:2.3.7" ], "ref_ids": [ 23, 31 ] } ], "ref_ids": [] }, { "id": 33, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.9", "categories": [], "title": "Intermediate Value Theorem for Derivatives", "contents": [ " Suppose that $f$ is differentiable on $[a,b],$ $f'(a)\\ne", "f'(b),$ and $\\mu$ is between $f'(a)$ and $f'(b).$ Then $f'(c)=\\mu$", "for some $c$ in $(a,b).$" ], "refs": [], "proofs": [ { "contents": [ "Suppose first that", "\\begin{equation}\\label{eq:2.3.17}", "f'(a)<\\mu0.", "\\end{equation}", "Since $g$ is", "continuous on $[a,b]$, $g$ attains a minimum at some point $c$ in", "$[a,b]$. Lemma~\\ref{thmtype:2.3.2} and \\eqref{eq:2.3.19} imply that there is a", "$\\delta>0$ such that", "$$", "g(x)0,\\quad f'(x)\\ge0,\\quad f'(x)<0,\\mbox{\\quad or\\quad} f'(x)", "\\le0,", "$$", "respectively$,$ for all $x$ in $(a,b).$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 38, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.14", "categories": [], "title": "", "contents": [ "If", "$$", "|f'(x)|\\le M,\\quad a0$. From \\eqref{eq:2.4.3}, there is an $x_0$ in $(a,b)$ such", "that", "\\begin{equation}\\label{eq:2.4.5}", "\\left|\\frac{f'(c)}{g'(c)}-L\\right|<\\epsilon\\mbox{\\quad if\\quad}", "x_0", "x_0$", "so that $f(x)\\ne0$ and $f(x)\\ne f(x_0)$ if $x_10.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "Since $f^{(r)}(x_0)=0$ for $1\\le r\\le n-1$,", " \\eqref{eq:2.5.7} implies that", "\\begin{equation}\\label{eq:2.5.10}", "f(x)-f(x_0)=\\left[\\frac{f^{(n)}(x_0)}{ n!}+E_n(x)\\right] (x-x_0)^n", "\\end{equation}", "in some interval containing $x_0$. Since $\\lim_{x\\to x_0} E_n(x)=0$", "and", "$f^{(n)}(x_0)\\ne0$, there is a $\\delta>0$ such that", "$$", "|E_n(x)|<\\left|\\frac{f^{(n)}(x_0)}{ n!}\\right|\\mbox{\\quad if\\quad}", "|x-x_0|", "<\\delta.", "$$", "\\newpage", "\\noindent", "This and \\eqref{eq:2.5.10} imply that", "\\begin{equation}\\label{eq:2.5.11}", "\\frac{f(x)-f(x_0)}{(x-x_0)^n}", "\\end{equation}", "has the same sign as $f^{(n)}(x_0)$ if $0<|x-x_0|<\\delta$. If $n$ is", "odd the denominator of \\eqref{eq:2.5.11} changes sign in every", "neighborhood of $x_0$, and therefore so must the numerator (since the", "ratio has constant sign for $0<|x-x_0|<\\delta$). Consequently,", "$f(x_0)$ cannot be a local extreme value of $f$. This proves \\part{a}. If", "$n$ is even, the denominator of \\eqref{eq:2.5.11} is positive for $x\\ne", "x_0$, so $f(x)-f(x_0)$ must have the same sign as", "$f^{(n)}(x_0)$ for $0<|x-x_0|<\\delta$. This proves \\part{b}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 42, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.5.4", "categories": [], "title": "Taylor's Theorem", "contents": [ "Suppose that $f^{(n+1)}$ exists on an open interval $I$ about $x_0,$", "and let", "$x$ be in $I.$ Then the remainder", "$$", "R_n(x)=f(x)-T_n(x)", "$$", "can be written as", "$$", "R_n(x)=\\frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1},", "$$", "where $c$ depends upon $x$ and is between $x$ and $x_0.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 43, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.5.5", "categories": [], "title": "Extended Mean Value Theorem", "contents": [ "Suppose that $f$ is continuous on a finite closed interval $I$ with", "endpoints $a$ and $b$ $($that is, either $I=(a,b)$ or $I=(b,a)),$", "$f^{(n+1)}$ exists on the open interval $I^0,$ and$,$ if $n>0,$ that", "$f'$, \\dots, $f^{(n)}$ exist and are continuous at $a.$ Then", "\\begin{equation}\\label{eq:2.5.17}", "f(b)-\\sum_{r=0}^n\\frac{f^{(r)}(a)}{ r!}(b-a)^r=\\frac{f^{(n+1)}(c)}{(n+1)!}", "(b-a)^{n+1}", "\\end{equation}", "for some $c$ in $I^0.$" ], "refs": [], "proofs": [ { "contents": [ "The proof is by induction. The mean value theorem", "(Theorem~\\ref{thmtype:2.3.11}) implies the conclusion for $n=0$.", "Now suppose that", "$n\\ge1$, and assume that the assertion of the theorem is true with $n$", "replaced by", "$n-1$. The left side of \\eqref{eq:2.5.17} can be written as", "\\begin{equation}\\label{eq:2.5.18}", "f(b)-\\sum_{r=0}^n\\frac{f^{(r)}(a)}{ r!}(b-a)^r=K\\frac{(b-a)^{n+1}}{(n+1)!}", "\\end{equation}", "for some number $K$. We must prove that $K=f^{(n+1)}(c)$ for", "some $c$ in $I^0$. To this end, consider the auxiliary function", "$$", "h(x)=f(x)-\\sum_{r=0}^n\\frac{f^{(r)}(a)}{", "r!}(x-a)^r-K\\frac{(x-a)^{n+1}}{", "(n+1)!},", "$$", "which satisfies", "$$", "h(a)=0,\\quad h(b)=0,", "$$", "(the latter because of \\eqref{eq:2.5.18}) and is continuous on the closed", "interval $I$ and differentiable on $I^0$, with", "\\begin{equation}\\label{eq:2.5.19}", "h'(x)=f'(x)-\\sum_{r=0}^{n-1}\\frac{f^{(r+1)}(a)}{", "r!}(x-a)^r-K\\frac{(x-a)^n}{n!}.", "\\end{equation}", "Therefore, Rolle's theorem (Theorem~\\ref{thmtype:2.3.8})", "implies that $h'(b_1)=0$ for some $b_1$ in", "$I^0$; thus, from \\eqref{eq:2.5.19},", "$$", "f'(b_1)-\\sum_{r=0}^{n-1}\\frac{f^{(r+1)}(a)}{", "r!}(b_1-a)^r-K\\frac{(b_1-a)^n}{n!}=0.", "$$", "If we temporarily write $f'=g$, this becomes", "\\begin{equation}\\label{eq:2.5.20}", "g(b_1)-\\sum_{r=0}^{n-1}\\frac{g^{(r)}(a)}{", "r~}(b_1-a)^r-K\\frac{(b_1-a)^n}{n!}=0.", "\\end{equation}", "\\newpage", "\\noindent", "Since $b_1\\in I^0$, the hypotheses on $f$ imply that $g$ is continuous", "on the closed interval $J$ with endpoints $a$ and $b_1$, $g^{(n)}$", "exists on", "$J^0$, and, if $n\\ge1$, $g'$, \\dots, $g^{(n-1)}$ exist and are", "continuous", "at $a$ (also at $b_1$, but this is not important). The induction", "hypothesis, applied to $g$ on the interval $J$, implies that", "$$", "g(b_1)-\\sum_{r=0}^{n-1}\\frac{g^{(r)}(a)}{ r!}", "(b_1-a)^r=\\frac{g^{(n)}(c)}{n!}(b_1-a)^n", "$$", "for some $c$ in $J^0$. Comparing this with \\eqref{eq:2.5.20} and recalling", "that $g=f'$ yields", "$$", "K=g^{(n)}(c)=f^{(n+1)}(c).", "$$", "Since $c$ is in $I^0$, this completes the induction." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.3.11", "TRENCH_REAL_ANALYSIS-thmtype:2.3.8" ], "ref_ids": [ 35, 32 ] } ], "ref_ids": [] }, { "id": 44, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.1.2", "categories": [], "title": "", "contents": [ "If $f$ is unbounded on $[a,b],$ then $f$ is not integrable on", "$[a,b].$" ], "refs": [], "proofs": [ { "contents": [ "We will show that if $f$ is unbounded on $[a,b]$, $P$ is any", "partition of $[a,b]$, and $M>0$, then there are Riemann sums $\\sigma$", "and $\\sigma'$ of $f$ over $P$ such that", "\\begin{equation} \\label{eq:3.1.7}", "|\\sigma-\\sigma'|\\ge M.", "\\end{equation}", "We leave it to you (Exercise~\\ref{exer:3.1.2}) to complete the proof by", "showing from this that", "$f$ cannot satisfy Definition~\\ref{thmtype:3.1.1}.", "Let", "$$", "\\sigma=\\sum_{j=1}^nf(c_j)(x_j-x_{j-1})", "$$", "be a Riemann sum of $f$ over a partition $P$ of $[a,b]$. There must be", "an integer $i$ in $\\{1,2, \\dots,n\\}$ such that", "\\begin{equation} \\label{eq:3.1.8}", "|f(c)-f(c_i)|\\ge \\frac{M }{ x_i-x_{i-1}}", "\\end{equation}", "for some $c$ in $[x_{i-1}x_i]$, because if there were not so, we", "would have", "$$", "|f(x)-f(c_j)|<\\frac{M}{ x_j-x_{j-1}},\\quad x_{j-1}\\le x\\le x_j,\\quad", "1\\le j\\le n.", "$$", "Then", "\\begin{eqnarray*}", "|f(x)|\\ar=|f(c_j)+f(x)-f(c_j)|\\le|f(c_j)|+|f(x)-f(c_j)|\\\\", "\\ar\\le |f(c_j)|+\\frac{M}{ x_j-x_{j-1}},\\quad x_{j-1}\\le x\\le x_j,\\quad", "1\\le j\\le n.", "\\end{eqnarray*}", "which implies that", "$$", "|f(x)|\\le\\max_{1\\le j\\le n}|f(c_j)|+\\frac{M}{", "x_j-x_{j-1}},", "\\quad a\\le x \\le b,", "$$", "contradicting the assumption that $f$ is unbounded on $[a,b]$.", " Now suppose that $c$ satisfies \\eqref{eq:3.1.8}, and", "consider the Riemann sum", "$$", "\\sigma'=\\sum_{j=1}^nf(c'_j)(x_j-x_{j-1})", "$$", "over the same partition $P$, where", "$$", "c'_j=\\left\\{\\casespace\\begin{array}{ll}", "c_j,&j \\ne i,\\\\", "c,&j=i.\\end{array}\\right.", "$$", "\\newpage", "\\noindent", "Since", "$$", "|\\sigma-\\sigma'|=|f(c)-f(c_i)|(x_i-x_{i-1}),", "$$", "\\eqref{eq:3.1.8} implies \\eqref{eq:3.1.7}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.1.1" ], "ref_ids": [ 315 ] } ], "ref_ids": [] }, { "id": 45, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.1.4", "categories": [], "title": "", "contents": [ "Let $f$ be bounded on $[a,b]$, and let $P$", "be a partition of $[a,b].$ Then", "\\begin{alist}", "\\item % (a)", " The upper sum $S(P)$ of $f$ over $P$ is the supremum", " of the set of all Riemann sums of $f$ over $P.$", "\\item % (b)", " The lower sum $s(P)$ of $f$ over $P$ is the infimum", " of the set of all Riemann sums of $f$ over $P.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "\\part{a} If $P=\\{x_0,x_1, \\dots,x_n\\}$, then", "$$", "S(P)=\\sum_{j=1}^n M_j(x_j-x_{j-1}),", "$$", "where", "$$", "M_j=\\sup_{x_{j-1}\\le x\\le x_j}f(x).", "$$", "An arbitrary Riemann sum of $f$ over $P$ is of the form", "$$", "\\sigma=\\sum_{j=1}^n f(c_j)(x_j-x_{j-1}),", "$$", "where $x_{j-1}\\le c_j\\le x_j$.", "Since $f(c_j)\\le M_j$, it follows that $\\sigma\\le S(P)$.", "Now let", "$\\epsilon>0$ and choose $\\overline c_j$ in $[x_{j-1},x_j]$ so that", "$$", "f(\\overline c_j) > M_j -\\frac{\\epsilon}{ n(x_j-x_{j-1})},\\quad 1\\le j\\le", "n.", "$$", "The Riemann sum produced in this way is", "$$", "\\overline \\sigma=\\sum_{j=1}^n", "f(\\overline", "c_j)(x_j-x_{j-1})>\\sum_{j=1}^n\\left[M_j-\\frac{\\epsilon}{", "n(x_j-x_{j-1})})\\right](x_j-x_{j-1})=S(P)-\\epsilon.", "$$", "Now Theorem~\\ref{thmtype:1.1.3} implies that", "$S(P)$ is the supremum of the set of Riemann sums of $f$", "over $P$.", "\\part{b} Exercise~\\ref{exer:3.1.7}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:1.1.3" ], "ref_ids": [ 1 ] } ], "ref_ids": [] }, { "id": 46, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.2", "categories": [], "title": "", "contents": [ "If $f$ is bounded on $[a,b],$ then", "\\begin{equation} \\label{eq:3.2.6}", "\\underline{\\int_a^b}f(x)\\,dx\\le\\overline{\\int_a^b}f(x)\\,dx.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $P_1$ and $P_2$ are partitions of $[a,b]$ and $P'$ is a", "refinement of both. Letting $P=P_1$ in \\eqref{eq:3.2.3} and $P=P_2$ in", "\\eqref{eq:3.2.2} shows that", "$$", "s(P_1)\\le s(P') \\mbox{\\quad and \\quad} S(P')\\le S(P_2).", "$$", "Since $s(P')\\le S(P')$, this implies that", "$s(P_1)\\le S(P_2)$.", "Thus, every lower sum is a lower bound for the set of all upper sums.", "Since $\\overline{\\int_a^b}f(x)\\,dx$ is the infimum of", "this set, it follows that", "$$", "s(P_1)\\le\\overline{\\int_a^b}f(x)\\,dx", "$$", "for every partition $P_1$ of $[a,b]$. This means that", "$\\overline{\\int_a^b}", "f(x)\\,dx$ is an upper bound for the set of all lower sums. Since", "$\\underline{\\int_a^b} f(x)\\,dx$ is the supremum of this set,", "this implies \\eqref{eq:3.2.6}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 47, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.3", "categories": [], "title": "", "contents": [ "If $f$ is integrable on $[a,b],$ then", "$$", "\\underline{\\int_a^b}f(x)\\,dx=\\overline{\\int_a^b}f(x)\\,dx=\\int_a^b", "f(x)\\,dx.", "$$" ], "refs": [], "proofs": [ { "contents": [ "We prove that", "$\\overline{\\int_a^b}f(x)\\,dx=\\int_a^bf(x)\\,dx$ and leave it to you to", "show that", "$\\underline{\\int_a^b}f(x)\\,dx=\\int_a^bf(x)\\,dx$", "(Exercise~\\ref{exer:3.2.2}).", " Suppose that $P$ is a partition of $[a,b]$", "and $\\sigma$ is a Riemann sum of $f$ over $P$.", "Since", "\\begin{eqnarray*}", "\\overline{\\int_a^b}f(x)\\,dx-\\int_a^b f(x)\\,dx\\ar=", "\\left(\\overline{\\int_a^b}f(x)\\,dx-S(P)\\right)+(S(P)-\\sigma)", "\\\\[2\\jot]", "&&+\\left(\\sigma-\\int_a^b f(x)\\ dx\\right),", "\\end{eqnarray*}", "\\newpage", "\\noindent", "the triangle inequality implies that", "\\begin{equation} \\label{eq:3.2.7}", "\\begin{array}{rcl}", "\\dst{\\left|\\overline{\\int_a^b}f(x)\\,dx-\\int_a^b f(x)\\,dx \\right|}\\ar\\le", "\\dst{\\left|\\overline{\\int_a^b}f(x)\\,dx-S(P)\\right|+|S(P)-\\sigma|}", "\\\\[2\\jot]", "&&+\\dst{\\left|\\sigma-\\int_a^b f(x)\\ dx\\right|}.", "\\end{array}", "\\end{equation}", "Now suppose that $\\epsilon>0$.", " From Definition~\\ref{thmtype:3.1.3}, there is", "a partition $P_0$ of $[a,b]$ such that", "\\begin{equation} \\label{eq:3.2.8}", "\\overline{\\int_a^b} f(x)\\,dx\\le S(P_0)<", "\\overline{\\int_a^b}f(x)\\,dx+\\frac{\\epsilon}{3}.", "\\end{equation}", "From Definition~\\ref{thmtype:3.1.1}, there is a $\\delta>0$ such that", "\\begin{equation} \\label{eq:3.2.9}", "\\left|\\sigma-\\int_a^bf(x)\\,dx\\right|<\\frac{\\epsilon}{3}", "\\end{equation}", "if $\\|P\\|<\\delta$. Now suppose that $\\|P\\|<\\delta$ and $P$ is a", "refinement of $P_0$. Since $S(P)\\le S(P_0)$ by Lemma~\\ref{thmtype:3.2.1},", "\\eqref{eq:3.2.8} implies that", "$$", "\\overline{\\int_a^b} f(x)\\,dx\\le S(P)<", "\\overline{\\int_a^b}f(x)\\,dx+\\frac{\\epsilon}{3},", "$$", "so", "\\begin{equation} \\label{eq:3.2.10}", "\\left|S(P)-\\overline{\\int_a^b}f(x)\\,dx\\right|<\\frac{\\epsilon}{3}", "\\end{equation}", "in addition to \\eqref{eq:3.2.9}. Now \\eqref{eq:3.2.7}, \\eqref{eq:3.2.9}, and", "\\eqref{eq:3.2.10} imply that", "\\begin{equation} \\label{eq:3.2.11}", "\\left|\\overline{\\int_a^b} f(x)\\,dx-\\int_a^b f(x)\\,dx\\right|<", "\\frac{2\\epsilon}{3}+|S(P)-\\sigma|", "\\end{equation}", "for every Riemann sum $\\sigma$ of $f$ over $P$. Since $S(P)$ is the", "supremum of these Riemann sums", "(Theorem~\\ref{thmtype:3.1.4}), we may choose", "$\\sigma$ so that", "$$", "|S(P)-\\sigma|<\\frac{\\epsilon}{3}.", "$$", "Now \\eqref{eq:3.2.11} implies that", "$$", "\\left|\\overline{\\int_a^b} f(x)\\,dx-\\int_a^b f(x)\\,dx \\right|<", "\\epsilon.", "$$", "Since $\\epsilon$ is an arbitrary positive number, it follows that", "$$", "\\overline{\\int_a^b}f(x)\\,dx=\\int_a^b f(x)\\,dx.", "$$", "\\vskip-6.5ex" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.1.3", "TRENCH_REAL_ANALYSIS-thmtype:3.1.1", "TRENCH_REAL_ANALYSIS-thmtype:3.2.1", "TRENCH_REAL_ANALYSIS-thmtype:3.1.4" ], "ref_ids": [ 316, 315, 246, 45 ] } ], "ref_ids": [] }, { "id": 48, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.5", "categories": [], "title": "", "contents": [ "If $f$ is bounded on $[a,b]$ and", "\\begin{equation} \\label{eq:3.2.16}", "\\underline{\\int_a^b} f(x)\\,dx=\\overline{\\int_a^b}f(x)\\,dx=L,", "\\end{equation}", "then $f$ is integrable on $[a,b]$ and", "\\begin{equation} \\label{eq:3.2.17}", "\\int_a^b f(x)\\,dx=L.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "If $\\epsilon>0$, there is a $\\delta>0$ such that", "\\begin{equation} \\label{eq:3.2.18}", "\\underline{\\int_a^b}f(x)\\,dx-\\epsilon0$ there is", "a partition $P$ of $[a,b]$ for which", "\\begin{equation} \\label{eq:3.2.19}", "S(P)-s(P)<\\epsilon.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "We leave it to you (Exercise~\\ref{exer:3.2.4}) to show that if $\\int_a^b", "f(x)\\,dx$ exists, then \\eqref{eq:3.2.19} holds for $\\|P\\|$ sufficiently", "small. This implies that the stated condition is necessary for", "integrability. To show that it is sufficient, we observe that since", "$$", "s(P)\\le \\underline{\\int_a^b}f(x)\\,dx\\le\\overline{\\int_a^b}f(x)\\,dx\\le", "S(P)", "$$", "for all $P$, \\eqref{eq:3.2.19} implies that", "$$", "0\\le\\overline{\\int_a^b} f(x)\\,dx-\\underline{\\int_a^b}f(x)\\,dx<", "\\epsilon.", "$$", "Since $\\epsilon$ can be any positive number, this implies that", "$$", "\\overline{\\int_a^b} f(x)\\,dx=\\underline{\\int_a^b} f(x)\\,dx.", "$$", "Therefore, $\\int_a^b f(x)\\,dx$ exists, by Theorem~\\ref{thmtype:3.2.5}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.2.5" ], "ref_ids": [ 48 ] } ], "ref_ids": [] }, { "id": 51, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.8", "categories": [], "title": "", "contents": [ "If $f$ is continuous on $[a,b],$", "then $f$ is integrable on $[a,b]$." ], "refs": [], "proofs": [ { "contents": [ "Let $P=\\{x_0,x_1, \\dots,x_n\\}$ be a partition of $[a,b]$. Since", "$f$ is continuous on $[a,b]$, there are points $c_j$ and $c'_j$ in", "$[x_{j-1},x_j]$ such that", "$$ f(c_j)=M_j=\\sup_{x_{j-1}\\le x\\le x_j}f(x)", "$$", "and", "$$", "f(c'_j)=m_j=\\inf_{x_{j-1}\\le x\\le x_j}f(x)", "$$", "(Theorem~\\ref{thmtype:2.2.9}).", "Therefore,", "\\begin{equation} \\label{eq:3.2.20}", "S(P)-s(P)=\\sum_{j=1}^n\\left[f(c_j)-f(c'_j)\\right](x_j-x_{j-1}).", "\\end{equation}", "Since $f$ is uniformly continuous on $[a,b]$", "(Theorem~\\ref{thmtype:2.2.12}), there is for each $\\epsilon>0$", "a", "$\\delta>0$ such that", " $$", "|f(x')-f(x)|<\\frac{\\epsilon}{ b-a}", " $$", " if $x$ and $x'$ are", "in $[a,b]$ and $|x-x'|<\\delta$. If $\\|P\\|<\\delta$, then", "$|c_j-c'_j|<\\delta$ and, from \\eqref{eq:3.2.20},", "$$", " S(P)-s(P)<\\frac{\\epsilon}{ b-a}", "\\sum_{j=1}^n(x_j-x_{j-1})=\\epsilon.", "$$", "Hence, $f$ is integrable", "on $[a,b]$, by Theorem~\\ref{thmtype:3.2.7}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.2.9", "TRENCH_REAL_ANALYSIS-thmtype:2.2.12", "TRENCH_REAL_ANALYSIS-thmtype:3.2.7" ], "ref_ids": [ 23, 25, 50 ] } ], "ref_ids": [] }, { "id": 52, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.9", "categories": [], "title": "", "contents": [ "If $f$ is monotonic on $[a,b],$ then $f$ is integrable on $[a,b]$." ], "refs": [], "proofs": [ { "contents": [ "Let $P=\\{x_0,x_1, \\dots,x_n\\}$ be a partition of $[a,b]$. Since", " $f$ is nondecreasing,", "\\begin{eqnarray*}", "f(x_j)\\ar=M_j=\\sup_{x_{j-1}\\le x\\le x_j}f(x)\\\\", "\\arraytext{and}\\\\", "f(x_{j-1})\\ar=m_j=\\inf_{x_{j-1}\\le x\\le x_j}f(x).", "\\end{eqnarray*}", "Hence,", "$$", "S(P)-s(P)=\\sum_{j=1}^n(f(x_j)-f(x_{j-1})) (x_j-x_{j-1}).", "$$", "Since $00$", "there are positive numbers $\\delta_1$ and $\\delta_2$ such that", "\\begin{eqnarray*}", "\\left|\\sigma_f-\\int_a^b f(x)\\,dx\\right|\\ar<\\frac{\\epsilon}{2}", "\\mbox{\\quad if\\quad}\\|P\\|<\\delta_1\\\\", "\\arraytext{and}\\\\", "\\left|\\sigma_g-\\int_a^b g(x)\\,dx\\right|\\ar<\\frac{\\epsilon}{2}", "\\mbox{\\quad if\\quad}\\|P\\|<\\delta_2.", "\\end{eqnarray*}", "If $\\|P\\|<\\delta=\\min(\\delta_1,\\delta_2)$, then", "\\begin{eqnarray*}", "\\left|\\sigma_{f+g}-\\int_a^b f(x)\\,dx-\\int_a^b g(x)\\,dx\\right|", "\\ar=\\left|\\left(\\sigma_f-\\int_a^b f(x)\\,dx\\right)+", "\\left(\\sigma_g-\\int_a^b g(x)\\,dx\\right)\\right|\\\\", "\\ar\\le \\left|\\sigma_f-\\int_a^b f(x)\\,dx\\right|+", "\\left|\\sigma_g-\\int_a^b g(x)\\,dx\\right|\\\\", "&<&\\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}=\\epsilon,", "\\end{eqnarray*}", "so the conclusion follows from Definition~\\ref{thmtype:3.1.1}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.1.1", "TRENCH_REAL_ANALYSIS-thmtype:3.1.1" ], "ref_ids": [ 315, 315 ] } ], "ref_ids": [] }, { "id": 54, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.2", "categories": [], "title": "", "contents": [ "If $f$ is integrable on $[a,b]$ and", "$c$ is a constant$,$ then $cf$ is integrable on $[a,b]$ and", "$$", "\\int_a^b cf(x)\\,dx=c\\int_a^b f(x)\\,dx.", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 55, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.3", "categories": [], "title": "", "contents": [ " If $f_1,$ $f_2,$ \\dots$,$ $f_n$ are", "integrable on $[a,b]$ and $c_1,$ $c_2,$ \\dots$,$ $c_n$ are", "constants$,$ then", "$c_1f_1+c_2f_2+\\cdots+ c_nf_n$ is integrable on $[a,b]$ and", "\\begin{eqnarray*}", "\\int_a^b (c_1f_1+c_2f_2+\\cdots+c_nf_n)(x)\\,dx\\ar=c_1\\int_a^b f_1(x)\\,dx", "+c_2\\int_a^b f_2(x)\\,dx\\\\", "\\ar{}+\\cdots+c_n\\int_a^b f_n(x)\\,dx.", "\\end{eqnarray*}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 56, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.4", "categories": [], "title": "", "contents": [ "If $f$ and $g$ are integrable on", "$[a,b]$ and $f(x)\\le g(x)$ for $a\\le x\\le b,$ then", "\\begin{equation}\\label{eq:3.3.1}", "\\int_a^b f(x)\\,dx\\le\\int_a^b g(x)\\,dx.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Since $g(x)-f(x)\\ge0$, every lower sum of $g-f$ over any", "partition of $[a,b]$ is nonnegative. Therefore,", "$$", "\\underline{\\int_a^b}(g(x)-f(x))\\,dx\\ge0.", "$$", "Hence,", "\\begin{equation}\\label{eq:3.3.2}", "\\begin{array}{rcl}", "\\dst\\int_a^b g(x)\\,dx-\\int_a^b f(x)\\,dx\\ar=\\dst\\int_a^b", "(g(x)-f(x))\\,dx\\\\[2\\jot]", "\\ar=\\dst\\underline{\\int_a^b}(g(x)-f(x))\\,dx\\ge0,", "\\end{array}", "\\end{equation}", "which yields \\eqref{eq:3.3.1}. (The first equality in \\eqref{eq:3.3.2}", "follows", "from Theorems~\\ref{thmtype:3.3.1} and \\ref{thmtype:3.3.2}; the second, from", "Theorem~\\ref{thmtype:3.2.3}.)" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.3.1", "TRENCH_REAL_ANALYSIS-thmtype:3.3.2", "TRENCH_REAL_ANALYSIS-thmtype:3.2.3" ], "ref_ids": [ 53, 54, 47 ] } ], "ref_ids": [] }, { "id": 57, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.5", "categories": [], "title": "", "contents": [ " If $f$ is integrable on $[a,b],$", "then so is $|f|$, and", "\\begin{equation} \\label{eq:3.3.3}", "\\left|\\int_a^b f(x)\\,dx\\right|\\le\\int_a^b |f(x)|\\,dx.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Let $P$ be a partition of $[a,b]$ and define", "\\begin{eqnarray*}", "M_j\\ar=\\sup\\set{f(x)}{x_{j-1}\\le x\\le x_j},\\\\", "m_j\\ar=", "\\inf\\set{f(x)}{x_{j-1}\\le x\\le x_j},\\\\", "\\overline{M}_j\\ar=\\sup\\set{|f(x)|}{x_{j-1}\\le x\\le x_j},\\\\", "\\overline{m}_j\\ar=\\inf\\set{|f(x)|}{x_{j-1}\\le x\\le x_j}.", "\\end{eqnarray*}", "Then", "\\begin{equation} \\label{eq:3.3.4}", "\\begin{array}{rcl}", "\\overline{M}_j-\\overline{m}_j\\ar=", "\\dst\\sup\\set{|f(x)|-|f(x')|}{x_{j-1}\\le x,x'\\le x_j}\\\\", "\\ar\\le \\dst\\sup\\set{|f(x)-f(x')|}{x_{j-1}\\le x,x'\\le x_j}\\\\", "\\ar=M_j-m_j.", "\\end{array}", "\\end{equation}", "Therefore,", "$$", "\\overline{S}(P)-\\overline{s}(P)\\le S(P)-s(P),", "$$", "where the upper and lower sums on the left are associated with $|f|$", "and those on the right are associated with $f$. Now suppose that", "$\\epsilon>0$. Since $f$ is integrable on $[a,b]$,", " Theorem~\\ref{thmtype:3.2.7} implies that", "there is a partition $P$ of $[a,b]$ such that $S(P)-s(P)<\\epsilon$.", "This inequality and \\eqref{eq:3.3.4} imply that $\\overline", "S(P)-\\overline s(P)<\\epsilon$.", " Therefore, $|f|$ is integrable on $[a,b]$,", " again by Theorem~\\ref{thmtype:3.2.7}.", "Since", "$$", "f(x)\\le|f(x)|\\mbox{\\quad and \\quad}-f(x)\\le|f(x)|,\\quad a\\le x\\le b,", "$$", "\\newpage", "\\noindent", " Theorems~\\ref{thmtype:3.3.2} and \\ref{thmtype:3.3.4} imply", "that", "$$", "\\int_a^b f(x)\\,dx\\le\\int_a^b|f(x)|\\,dx\\mbox{\\quad and }", "-\\int_a^b f(x)\\,dx\\le\\int_a^b|f(x)|\\,dx,", "$$", "which implies \\eqref{eq:3.3.3}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.2.7", "TRENCH_REAL_ANALYSIS-thmtype:3.2.7", "TRENCH_REAL_ANALYSIS-thmtype:3.3.2", "TRENCH_REAL_ANALYSIS-thmtype:3.3.4" ], "ref_ids": [ 50, 50, 54, 56 ] } ], "ref_ids": [] }, { "id": 58, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.6", "categories": [], "title": "", "contents": [ "If $f$ and $g$ are integrable on $[a,b],$ then so is the product", "$fg.$" ], "refs": [], "proofs": [ { "contents": [ "We consider the case where $f$ and $g$ are nonnegative, and", "leave the rest of the proof to you (Exercise~\\ref{exer:3.3.4}). The", "subscripts $f$, $g$, and $fg$ in the following argument identify the", "functions", "with which the various quantities are associated. We assume that", "neither $f$ nor $g$ is identically zero on $[a,b]$, since the", "conclusion is obvious if one of them is.", "If $P=\\{x_0,x_1, \\dots,x_n\\}$ is a partition of $[a,b]$, then", "\\begin{equation}\\label{eq:3.3.5}", "S_{fg}(P)-s_{fg}(p)=\\sum_{j=1}^n (M_{fg,j}-m_{fg,", "j})(x_j-x_{j-1}).", "\\end{equation}", "Since $f$ and $g$ are nonnegative, $M_{fg,j}\\le M_{f,j}M_{g,j}$ and", "$m_{fg,j}\\ge m_{f,j}m_{g,j}$. Hence,", "\\begin{eqnarray*}", "M_{fg,j}-m_{fg,j}\\ar\\le M_{f,j}M_{g,j}-m_{f,", "j}m_{g,j}\\\\[2\\jot]", "\\ar=(M_{f,j}-m_{f,j})M_{g,j}+m_{f,j}(M_{g,j}-", "m_{g,j})\\\\[2\\jot]", "\\ar\\le M_g(M_{f,j}-m_{f,j})+M_f(M_{g,j}-m_{g,j}),", "\\end{eqnarray*}", "where $M_f$ and $M_g$ are upper bounds for $f$ and $g$ on $[a,b]$. From", "\\eqref{eq:3.3.5} and the last inequality,", "\\begin{equation} \\label{eq:3.3.6}", "S_{fg}(P)-s_{fg}(P)\\le M_g[S_f(P)-s_f(P)]+M_f[S_g(P)-s_g(P)].", "\\end{equation}", "Now suppose that $\\epsilon>0$. Theorem~\\ref{thmtype:3.2.7}", "implies that there are partitions $P_1$ and $P_2$ of $[a,b]$ such that", "\\begin{equation} \\label{eq:3.3.7}", "S_f(P_1)-s_f(P_1)<\\frac{\\epsilon}{2M_g}\\mbox{\\quad and\\quad}", "S_g(P_2)-s_g(P_2)<\\frac{\\epsilon}{2M_f}.", "\\end{equation}", "If $P$ is a refinement of both $P_1$ and $P_2$,", " then \\eqref{eq:3.3.7}", "and Lemma~\\ref{thmtype:3.2.1} imply that", "$$", "S_f(P)-s_f(P)<\\frac{\\epsilon}{2M_g}\\mbox{\\quad and\\quad}", "S_g(P)-s_g(P)<\\frac{\\epsilon}{2M_f}.", "$$", "This and \\eqref{eq:3.3.6} yield", "$$", "S_{fg}(P)-s_{fg}(P)<\\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}=\\epsilon.", "$$", " Therefore, $fg$ is integrable on $[a,b]$, by", "Theorem~\\ref{thmtype:3.2.7}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.2.7", "TRENCH_REAL_ANALYSIS-thmtype:3.2.1", "TRENCH_REAL_ANALYSIS-thmtype:3.2.7" ], "ref_ids": [ 50, 246, 50 ] } ], "ref_ids": [] }, { "id": 59, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.7", "categories": [], "title": "First Mean Value Theorem for Integrals", "contents": [ "Suppose that $u$ is continuous and $v$ is integrable and nonnegative", "on", "$[a,b].$ Then", "\\begin{equation} \\label{eq:3.3.8}", "\\int_a^b u(x)v(x)\\,dx=u(c)\\int_a^b v(x)\\,dx", "\\end{equation}", "for some $c$ in $[a,b]$." ], "refs": [], "proofs": [ { "contents": [ "From Theorem~\\ref{thmtype:3.2.8}, $u$ is integrable on", "$[a,b]$. Therefore,", "Theorem~\\ref{thmtype:3.3.6} implies", "that the integral on the left exists. If $m=\\min\\set{u(x)}{a\\le x\\le", "b}$", " and $M=\\max\\set{u(x)}{a\\le x\\le b}$ (recall", "Theorem~\\ref{thmtype:2.2.9}), then", "$$", "m\\le u(x)\\le M", "$$", "and, since $v(x)\\ge0$,", "$$", "mv(x)\\le u(x) v(x)\\le Mv(x).", "$$", "Therefore, Theorems~\\ref{thmtype:3.3.2} and", "\\ref{thmtype:3.3.4} imply that", "\\vskip2pt", "\\begin{equation} \\label{eq:3.3.9}", "m\\int_a^b v(x)\\,dx\\le\\int_a^b u(x)v(x)\\,dx\\le M\\int_a^b v(x)\\,dx.", "\\end{equation}", "\\vskip2pt", "This implies that \\eqref{eq:3.3.8} holds for any $c$ in $[a,b]$", "if $\\int_a^b v(x)\\,dx=0$. If $\\int_a^b v(x)\\,dx\\ne0$, let", "\\vskip1pt", "\\begin{equation} \\label{eq:3.3.10}", "\\overline{u}=\\frac{\\dst\\int_a^b u(x)v(x)\\,dx}{\\dst\\int_a^bv(x)\\,dx}", "\\end{equation}", "\\vskip1pt", "\\noindent Since $\\int_a^b v(x)\\,dx>0$ in this case (why?),", "\\eqref{eq:3.3.9} implies", "that $m\\le\\overline{u}\\le M$, and the intermediate value theorem", " (Theorem~\\ref{thmtype:2.2.10}) implies that $\\overline{u}=u(c)$", "for some $c$ in $[a,b]$. This implies \\eqref{eq:3.3.8}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.2.8", "TRENCH_REAL_ANALYSIS-thmtype:3.3.6", "TRENCH_REAL_ANALYSIS-thmtype:2.2.9", "TRENCH_REAL_ANALYSIS-thmtype:3.3.2", "TRENCH_REAL_ANALYSIS-thmtype:3.3.4", "TRENCH_REAL_ANALYSIS-thmtype:2.2.10" ], "ref_ids": [ 51, 58, 23, 54, 56, 24 ] } ], "ref_ids": [] }, { "id": 60, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.8", "categories": [], "title": "", "contents": [ "If $f$ is integrable on $[a,b]$", "and $a\\le a_10$. From Theorem~\\ref{thmtype:3.2.7},", "there is a partition $P=\\{x_0,x_1, \\dots,x_n\\}$ of $[a,b]$ such that", "\\begin{equation} \\label{eq:3.3.11}", "S(P)-s(P)=\\sum_{j=1}^n(M_j-m_j)(x_j-x_{j-1})<\\epsilon.", "\\end{equation}", "We may assume that $a_1$ and $b_1$ are partition points of $P$,", "because if not they can be inserted to obtain a refinement", "$P'$ such that $S(P')-s(P')\\le S(P)-s(P)$", "(Lemma~\\ref{thmtype:3.2.1}). Let", "$a_1=x_r$ and $b_1=x_s$. Since every term in \\eqref{eq:3.3.11} is", "nonnegative,", "$$", "\\sum_{j=r+1}^s (M_j-m_j)(x_j-x_{j-1})<\\epsilon.", "$$", "Thus, $\\overline{P}=\\{x_r,x_{r+1}, \\dots,x_s\\}$ is a partition of", "$[a_1,b_1]$ over which the upper and lower sums of $f$ satisfy", "$$", "S(\\overline{P})-s(\\overline{P})<\\epsilon.", "$$", " Therefore, $f$ is integrable on $[a_1,b_1]$, by", "Theorem~\\ref{thmtype:3.2.7}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.2.7", "TRENCH_REAL_ANALYSIS-thmtype:3.2.1", "TRENCH_REAL_ANALYSIS-thmtype:3.2.7" ], "ref_ids": [ 50, 246, 50 ] } ], "ref_ids": [] }, { "id": 61, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.9", "categories": [], "title": "", "contents": [ "If $f$ is integrable on $[a,b]$", "and $[b,c],$ then $f$ is integrable on $[a,c],$ and", "\\begin{equation} \\label{eq:3.3.12}", "\\int_a^cf(x)\\,dx=\\int_a^bf(x)\\,dx+\\int_b^cf(x)\\,dx.", "\\end{equation}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 62, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.10", "categories": [], "title": "", "contents": [ "If $f$ is integrable on $[a,b]$ and", "$a\\le c\\le b,$ then the function", "$F$ defined by", "$$", " F(x)=\\int_c^x f(t)\\,dt", "$$", " satisfies a Lipschitz", "condition on $[a,b],$ and is therefore", "continuous on", "$[a,b].$" ], "refs": [], "proofs": [ { "contents": [ "If $x$ and $x'$ are in $[a,b]$, then", "$$", "F(x)-F(x')=\\int_c^x f(t)\\,dt-\\int_c^{x'} f(t)\\,dt=\\int_{x'}^x f(t)\\,", "dt,", "$$", "by Theorem~\\ref{thmtype:3.3.9} and the conventions just adopted. Since", "$|f(t)|\\le K$ $(a\\le t\\le b)$ for some constant $K$,", "$$", "\\left|\\int_{x'}^x f(t)\\,dt\\right|\\le K|x-x'|,\\quad a\\le x,\\, x'\\le b", "$$", "(Theorem~\\ref{thmtype:3.3.5}), so", "$$", "|F(x)-F(x')|\\le K|x-x'|,\\quad a\\le x,\\,x'\\le b.", "$$", "\\vskip-2em" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.3.9", "TRENCH_REAL_ANALYSIS-thmtype:3.3.5" ], "ref_ids": [ 61, 57 ] } ], "ref_ids": [] }, { "id": 63, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.11", "categories": [], "title": "", "contents": [ "If $f$ is integrable on $[a,b]$ and $a\\le c\\le b,$ then", "$F(x)=\\int_c^x", "f(t)\\,dt$ is differentiable at any point $x_0$ in $(a,b)$ where $f$ is", "continuous$,$ with $F'(x_0)=f(x_0).$ If $f$ is continuous from the", "right at $a,$ then $F_+'(a)=f(a)$. If $f$ is continuous from", "the left at $b,$ then $F_-'(b)=f(b).$" ], "refs": [], "proofs": [ { "contents": [ "We consider the case where $a0$ a", "$\\delta>0$ such that", "$$", "|f(t)-f(x_0)|<\\epsilon\\mbox{\\quad if\\quad} |x-x_0|<\\delta", "$$", "and $t$ is between $x$ and $x_0$. Therefore, from \\eqref{eq:3.3.13},", "$$", "\\left|\\frac{F(x)-F(x_0)}{ x-x_0}-f(x_0)\\right|<\\epsilon", "\\frac{|x-x_0|}{", "|x-x_0|}=\\epsilon\\mbox{\\quad if\\quad} 0<|x-x_0|<\\delta.", "$$", "Hence, $F'(x_0)=f(x_0)$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.3.5" ], "ref_ids": [ 57 ] } ], "ref_ids": [] }, { "id": 64, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.12", "categories": [], "title": "", "contents": [ "Suppose that $F$ is continuous on the closed interval $[a,b]$ and", "differentiable on the open interval", "$(a,b),$ and $f$ is integrable on $[a,b].$ Suppose also that", "$$", "F'(x)=f(x),\\quad a0$ a $\\delta>0$", "such that", "$$", "\\left|\\sigma-\\int_a^b f(x)\\,dx\\right|<\\epsilon\\mbox{\\quad if\\quad}", "\\|P\\|<\\delta.", "$$", "Therefore,", "$$", "\\left|F(b)-F(a)-\\int_a^b f(x)\\,dx\\right|<\\epsilon", "$$", "for every $\\epsilon>0$, which implies \\eqref{eq:3.3.14}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.3.11" ], "ref_ids": [ 35 ] } ], "ref_ids": [] }, { "id": 65, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.14", "categories": [], "title": "Fundamental Theorem of Calculus", "contents": [ "If $f$ is continuous on $[a,b],$ then $f$ has an antiderivative on", "$[a,b].$ Moreover$,$ if $F$ is any antiderivative of $f$ on $[a,b],$", "then", "$$", "\\int_a^b f(x)\\,dx=F(b)-F(a).", "$$" ], "refs": [], "proofs": [ { "contents": [ "The function", " $F_0(x)=\\int_a^x f(t)\\,dt$ is", "continuous on $[a,b]$ by Theorem~\\ref{thmtype:3.3.10}, and $F_0'(x)", "=f(x)$ on $(a,b)$ by Theorem~\\ref{thmtype:3.3.11}. Therefore,", "$F_0$ is an antiderivative of $f$ on $[a,b]$.", "Now let $F=F_0+c$ ($c=$ constant) be an arbitrary antiderivative of", "$f$ on $[a,b]$. Then", "\\vskip-2pt", "$$", "F(b)-F(a)=\\int_a^b f(x)\\,dx+c-\\int_a^a f(x)\\,dx-c=\\int_a^b f(x)\\,dx.", "$$", "\\vskip-2.5em" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.3.10", "TRENCH_REAL_ANALYSIS-thmtype:3.3.11" ], "ref_ids": [ 62, 63 ] } ], "ref_ids": [] }, { "id": 66, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.15", "categories": [], "title": "Integration by Parts", "contents": [ "If $u'$ and $v'$ are integrable on $[a,b],$ then", "\\begin{equation}\\label{eq:3.3.16}", "\\int_a^b u(x)v'(x)\\,dx=u(x)v(x)\\bigg|^b_a-\\int_a^b v(x)u'(x)\\,dx.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Since $u$ and $v$ are continuous", "on", "$[a,b]$ (Theorem~\\ref{thmtype:2.3.3}), they", "are integrable on $[a,b]$. Therefore, Theorems~\\ref{thmtype:3.3.1} and", "\\ref{thmtype:3.3.6} imply that the function", "$$", "(uv)'=u'v+uv'", "$$", "is integrable on $[a,b]$, and Theorem~\\ref{thmtype:3.3.12} implies that", "$$", "\\int_a^b[u(x)v'(x)+u'(x)v(x)]\\,dx=u(x)v(x)\\bigg|^b_a,", "$$", "which implies \\eqref{eq:3.3.16}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.3.3", "TRENCH_REAL_ANALYSIS-thmtype:3.3.1", "TRENCH_REAL_ANALYSIS-thmtype:3.3.6", "TRENCH_REAL_ANALYSIS-thmtype:3.3.12" ], "ref_ids": [ 28, 53, 58, 64 ] } ], "ref_ids": [] }, { "id": 67, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.16", "categories": [], "title": "Second Mean Value Theorem for Integrals", "contents": [ "Suppose that $f'$ is nonnegative and integrable and $g$ is", "continuous on $[a,b].$ Then", "\\begin{equation}\\label{eq:3.3.17}", "\\int_a^b f(x)g(x)\\,dx=f(a)\\int_a^c g(x)\\,dx+f(b)\\int_c^b g(x)\\,dx", "\\end{equation}", "for some $c$ in $[a,b].$" ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is differentiable on $[a,b]$, it is continuous on $[a,b]$", "(Theorem~\\ref{thmtype:2.3.3}).", "Since $g$ is continuous on $[a,b]$, so is $fg$", "(Theorem~\\ref{thmtype:2.2.5}). Therefore,", "Theorem~\\ref{thmtype:3.2.8} implies", "that the integrals in \\eqref{eq:3.3.17} exist. If", "\\begin{equation}\\label{eq:3.3.18}", "G(x)=\\int_a^x g(t)\\,dt,", "\\end{equation}", "then $G'(x)=g(x),\\ a0$ and $f(x)\\ge0$ on some subinterval", "$[a_1,b)$ of $[a,b),$ and", "\\begin{equation}\\label{eq:3.4.3}", "\\lim_{x\\to b-}\\frac{f(x)}{ g(x)}=M.", "\\end{equation}", "\\begin{alist}", "\\item % (a)", "If $00$. Then", "$$", "W_f[x_0-h,x_0+h]<\\epsilon", "$$", "for some $h>0$, so", "$$", "|f(x)-f(x')|<\\epsilon\\mbox{\\quad if\\quad} x_0-h\\le x,x'\\le x_0+h.", "$$", " Letting $x'=x_0$, we conclude that", "$$", "|f(x)-f(x_0)|<\\epsilon\\mbox{\\quad if\\quad} |x-x_0|0$, there is a", "$\\delta>0$ such that", "$$", "|f(x)-f(x_0)|<\\frac{\\epsilon}{2}\\mbox{\\quad and\\quad} |f(x')-f(x_0)|<", "\\frac{\\epsilon}{2}", "$$", "if $x_0-\\delta\\le x$, $x'\\le x_0+\\delta$. From the triangle", "inequality,", "$$", "|f(x)-f(x')|\\le|f(x)-f(x_0)|+|f(x')-f(x_0)|<\\epsilon,", "$$", "so", "$$", "W_f[x_0-h,x_0+h]\\le\\epsilon\\mbox{\\quad if\\quad} h<\\delta;", "$$", " therefore, $w_f(x_0)=0$.", "Similar arguments apply if", "$x_0=a$ or $x_0=b$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 80, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.5.6", "categories": [], "title": "", "contents": [ "A bounded function $f$ is integrable on a finite interval $[a,b]$ if", "and only if the set $S$ of discontinuities of $f$ in $[a,b]$ is of", "Lebesgue measure zero$.$" ], "refs": [], "proofs": [ { "contents": [ "From Theorem~\\ref{thmtype:3.5.2},", "$$", "S=\\set{x\\in [a,b]}{w_f(x)>0}\\negthickspace.", "$$", "Since $w_f(x)>0$ if and only if $w_f(x)\\ge1/i$ for some positive", "integer $i$, we can write", "\\begin{equation} \\label{eq:3.5.12}", "S=\\bigcup^\\infty_{i=1} S_i,", "\\end{equation}", "where", "$$", "S_i=\\set{x\\in [a,b]}{w_f(x)\\ge1/i}.", "$$", "Now suppose that $f$ is integrable on $[a,b]$ and $\\epsilon>0$.", "From Lemma~\\ref{thmtype:3.5.4},", " each $S_i$ can be covered by a finite number of", "open intervals $I_{i1}$, $I_{i2}$, \\dots, $I_{in}$ of total length", "less than", "$\\epsilon/2^i$. We simply renumber these intervals consecutively;", "thus,", "$$", "I_1,I_2, \\dots=", "I_{11}, \\dots,I_{1n_1},I_{21}, \\dots,I_{2n_2}, \\dots,", "I_{i1}, \\dots,I_{in_i}, \\dots.", "$$", "Now \\eqref{eq:3.5.8} and \\eqref{eq:3.5.9} hold because of \\eqref{eq:3.5.11} and", "\\eqref{eq:3.5.12}, and we have shown that the stated condition is", "necessary for integrability.", "For sufficiency, suppose that the stated condition holds and", "$\\epsilon>0$. Then $S$ can be covered by open intervals", "$I_1,I_2, \\dots$ that satisfy \\eqref{eq:3.5.9}. If $\\rho>0$, then the", "set", "$$", "E_\\rho=\\set{x\\in [a,b]}{w_f(x)\\ge\\rho}", "$$", "of Lemma~\\ref{thmtype:3.5.4} is contained in $S$", "(Theorem~\\ref{thmtype:3.5.2}), and therefore $E_\\rho$ is covered by", "$I_1,I_2, \\dots$. Since $E_\\rho$ is closed (Lemma~\\ref{thmtype:3.5.4})", "and bounded, the Heine--Borel theorem implies that $E_\\rho$ is covered", "by a finite number of intervals from $I_1,I_2, \\dots$. The sum of", "the lengths of the latter is less than $\\epsilon$, so", "Lemma~\\ref{thmtype:3.5.4} implies that $f$ is integrable on $[a,b]$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.5.2", "TRENCH_REAL_ANALYSIS-thmtype:3.5.4", "TRENCH_REAL_ANALYSIS-thmtype:3.5.4", "TRENCH_REAL_ANALYSIS-thmtype:3.5.2", "TRENCH_REAL_ANALYSIS-thmtype:3.5.4", "TRENCH_REAL_ANALYSIS-thmtype:3.5.4" ], "ref_ids": [ 79, 249, 249, 79, 249, 249 ] } ], "ref_ids": [] }, { "id": 81, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.2", "categories": [], "title": "", "contents": [ "The limit of a convergent sequence is unique$.$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that", "$$", "\\lim_{n\\to\\infty}s_n=s\\mbox{\\quad and \\quad}", "\\lim_{n\\to\\infty}s_n=s'.", "$$", "\\vskip5pt", "\\noindent We must show that $s=s'$.", "Let $\\epsilon>0$. From Definition~\\ref{thmtype:4.1.1}, there are", "integers $N_1$ and $N_2$ such that", "$$", "|s_n-s|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N_1", "$$", "\\vskip5pt", "\\noindent(because $\\lim_{n\\to\\infty} s_n=s$), and", "$$", "|s_n-s'|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N_2", "$$", "\\newpage", "\\noindent", "(because $\\lim_{n\\to\\infty}s_n=s'$). These inequalities both hold if", "$n\\ge N=\\max (N_1,N_2)$, which implies that", "\\begin{eqnarray*}", "|s-s'|\\ar=|(s-s_N)+(s_N-s')|\\\\", "\\ar\\le |s-s_N|+|s_N-s'|<\\epsilon+\\epsilon=2\\epsilon.", "\\end{eqnarray*}", "Since this inequality holds for every $\\epsilon>0$ and $|s-s'|$", "is independent of $\\epsilon$, we conclude that $|s-s'|=0$; that is,", "$s=s'$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.1.1" ], "ref_ids": [ 324 ] } ], "ref_ids": [] }, { "id": 82, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.4", "categories": [], "title": "", "contents": [ "A convergent sequence is bounded$.$" ], "refs": [], "proofs": [ { "contents": [ "By taking $\\epsilon=1$ in \\eqref{eq:4.1.2}, we see that if", " $\\lim_{n\\to\\infty} s_n=s$, then there is an integer $N$", "such that", "$$", "|s_n-s|<1\\mbox{\\quad if\\quad} n\\ge N.", "$$", "Therefore,", "$$", "|s_n|=|(s_n-s)+s|\\le|s_n-s|+|s|<1+|s|\\mbox{\\quad if\\quad} n\\ge N,", "$$", "and", "$$", "|s_n|\\le\\max\\{|s_0|,|s_1|, \\dots,|s_{N-1}|, 1+|s|\\}", "$$", "for all $n$, so $\\{s_n\\}$ is bounded." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 83, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.6", "categories": [], "title": "", "contents": [ "\\begin{alist}", "\\item % (a)", " If $\\{s_n\\}$ is nondecreasing$,$", "then $\\lim_{n\\to\\infty}s_n=\\sup\\{s_n\\}.$", "\\item % (b", "If $\\{s_n\\}$ is nonincreasing$,$ then $\\lim_{n\\to\\infty}s_n=", "\\inf\\{s_n\\}.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "\\part{a}. Let $\\beta=\\sup\\{s_n\\}$.", "If $\\beta<\\infty$, Theorem~\\ref{thmtype:1.1.3}", "implies that if $\\epsilon>0$ then", "$$", "\\beta-\\epsilonb$", "for some integer $N$. Then $s_n>b$ for $n\\ge N$, so", "$\\lim_{n\\to\\infty}s_n=\\infty$.", "We leave the proof of \\part{b}", "to you (Exercise~\\ref{exer:4.1.8})" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:1.1.3", "TRENCH_REAL_ANALYSIS-thmtype:4.1.1" ], "ref_ids": [ 1, 324 ] } ], "ref_ids": [] }, { "id": 84, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.7", "categories": [], "title": "", "contents": [ " Let $\\lim_{x\\to\\infty} f(x)=L,$", "where $L$ is in the extended reals$,$ and suppose that", "$s_n=f(n)$ for large $n.$ Then", "$$", "\\lim_{n\\to\\infty}s_n=L.", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 85, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.8", "categories": [], "title": "", "contents": [ " Let", "\\begin{equation}\\label{eq:4.1.4}", "\\lim_{n\\to\\infty} s_n=s\\mbox{\\quad and\\quad}\\lim_{n\\to\\infty} t_n=t,", "\\end{equation}", "where $s$ and $t$ are finite$.$ Then", "\\begin{equation}\\label{eq:4.1.5}", "\\lim_{n\\to\\infty} (cs_n)=cs", "\\end{equation}", "if $c$ is a constant$;$", "\\begin{eqnarray}", "\\lim_{n\\to\\infty}(s_n+t_n)\\ar=s+t,\\label{eq:4.1.6}\\\\", "\\lim_{n\\to\\infty}(s_n-t_n)\\ar=s-t, \\label{eq:4.1.7}\\\\", "\\lim_{n\\to\\infty}(s_nt_n)\\ar=st,\\label{eq:4.1.8}\\\\", "\\arraytext{and}\\nonumber\\\\", "\\lim_{n\\to\\infty}\\frac{s_n}{ t_n}\\ar=\\frac{s}{ t}\\label{eq:4.1.9}", "\\end{eqnarray}", "if $t_n$ is nonzero for all $n$ and $t\\ne0$." ], "refs": [], "proofs": [ { "contents": [ "We prove \\eqref{eq:4.1.8} and \\eqref{eq:4.1.9}", "and leave the rest to you", "(Exercises~\\ref{exer:4.1.15} and \\ref{exer:4.1.17}). For", "\\eqref{eq:4.1.8}, we write", "$$", "s_nt_n-st=s_nt_n-st_n+st_n-st", "=(s_n-s)t_n+s(t_n-t);", "$$", "\\newpage", "\\noindent", "hence,", "\\begin{equation}\\label{eq:4.1.10}", "|s_nt_n-st|\\le |s_n-s|\\,|t_n|+|s|\\,|t_n-t|.", "\\end{equation}", "Since $\\{t_n\\}$ converges, it is bounded (Theorem~\\ref{thmtype:4.1.4}).", "Therefore, there is a number $R$ such that $|t_n|\\le R$ for all $n$,", "and", "\\eqref{eq:4.1.10} implies that", "\\begin{equation}\\label{eq:4.1.11}", "|s_nt_n-st|\\le R|s_n-s|+|s|\\,|t_n-t|.", "\\end{equation}", "From \\eqref{eq:4.1.4}, if $\\epsilon>0$ there are integers", "$N_1$ and $N_2$ such that", "\\begin{eqnarray}", "|s_n-s|\\ar<\\epsilon\\mbox{\\quad if\\quad} n\\ge N_1 \\label{eq:4.1.12}\\\\", "\\arraytext{and}\\nonumber\\\\", "|t_n-t|\\ar<\\epsilon\\mbox{\\quad if\\quad} n\\ge N_2.\\label{eq:4.1.13}", "\\end{eqnarray}", "If $N=\\max (N_1,N_2)$, then \\eqref{eq:4.1.12} and \\eqref{eq:4.1.13} both hold", "when $n\\ge N$, and \\eqref{eq:4.1.11} implies that", "$$", "|s_nt_n-st|\\le (R+|s|)\\epsilon\\mbox{\\quad if\\quad} n\\ge N.", "$$", "This proves \\eqref{eq:4.1.8}.", "Now consider \\eqref{eq:4.1.9} in the special case where $s_n=1$ for all", "$n$ and $t\\ne 0$; thus, we want to show that", "$$", "\\lim_{n\\to\\infty}\\frac{1}{ t_n}=\\frac{1}{ t}.", "$$", "First, observe that since $\\lim_{n\\to\\infty} t_n=t\\ne0$, there is an", "integer $M$ such that $|t_n|\\ge |t|/2$ if $n\\ge M$. To see this,", "we apply Definition~\\ref{thmtype:4.1.1} with $\\epsilon=|t|/2$; thus,", "there is an integer $M$ such that $|t_n-t|<|t/2|$ if $n\\ge M$.", "Therefore,", "$$", "|t_n|=|t+(t_n-t)|\\ge ||t|-|t_n-t||\\ge\\frac{|t|}{2}\\mbox{\\quad if", "\\quad} n\\ge M.", "$$", " If $\\epsilon>0$, choose $N_0$ so that $|t_n-t|<\\epsilon$", "if $n\\ge N_0$,", " and let $N=\\max (N_0,M)$. Then", "$$", "\\left|\\frac{1}{ t_n}-\\frac{1}{ t}\\right|=\\frac{|t-t_n|}{", "|t_n|\\,|t|}\\le\\frac {2", "\\epsilon}{ |t|^2}\\mbox{\\quad if\\quad} n\\ge N;", "$$", "hence, $\\lim_{n\\to\\infty} 1/t_n=1/t$.", "Now we obtain \\eqref{eq:4.1.9} in the general case from \\eqref{eq:4.1.8}", "with $\\{t_n\\}$ replaced by $\\{1/t_n\\}$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.1.4", "TRENCH_REAL_ANALYSIS-thmtype:4.1.1" ], "ref_ids": [ 82, 324 ] } ], "ref_ids": [] }, { "id": 86, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.9", "categories": [], "title": "", "contents": [ "\\begin{alist}", "\\item % (a)", "If $\\{s_n\\}$ is bounded above and does not diverge to $-\\infty,$ then", "there is a unique real number $\\overline{s}$ such that$,$ if", "$\\epsilon>0,$", "\\begin{equation}\\label{eq:4.1.16}", "s_n<\\overline{s}+\\epsilon\\mbox{\\quad for large $n$}", "\\end{equation}", "and", "\\begin{equation}\\label{eq:4.1.17}", "s_n>\\overline{s}-\\epsilon\\mbox{\\quad for infinitely many", " $n$}.", "\\end{equation}", "\\item % (b)", "If $\\{s_n\\}$ is bounded below and does not diverge to $\\infty,$ then", "there is a unique real number $\\underline{s}$ such that$,$ if", "$\\epsilon", ">0,$", "\\begin{equation}\\label{eq:4.1.18}", "s_n>\\underline{s}-\\epsilon\\mbox{\\quad for large $n$}", "\\end{equation}", "and", "\\begin{equation}\\label{eq:4.1.19}", "s_n<\\underline{s}+\\epsilon\\mbox{\\quad for infinitely many", "$n$}.", "\\end{equation}", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "We will prove \\part{a} and leave the proof of \\part{b} to you", "(Exercise~\\ref{exer:4.1.23}). Since $\\{s_n\\}$ is bounded above,", "there is a number $\\beta$ such that $s_n<\\beta$ for all", "$n$. Since $\\{s_n\\}$ does not diverge to $-\\infty$, there is", "a number $\\alpha$ such that", "$s_n> \\alpha$ for infinitely many $n$. If we define", "$$", "M_k=\\sup\\{s_k,s_{k+1}, \\dots,s_{k+r}, \\dots\\},", "$$", "\\newpage", "\\noindent", "then $\\alpha\\le M_k\\le\\beta$, so $\\{M_k\\}$ is bounded. Since", "$\\{M_k\\}$ is nonincreasing (why?), it converges, by", "Theorem~\\ref{thmtype:4.1.6}. Let", "\\begin{equation} \\label{eq:4.1.20}", "\\overline{s}=\\lim_{k\\to\\infty} M_k.", "\\end{equation}", "If $\\epsilon>0$, then $M_k<\\overline{s}+\\epsilon$ for large $k$, and", "since $s_n\\le M_k$ for $n\\ge k$, $\\overline{s}$ satisfies", "\\eqref{eq:4.1.16}.", "If \\eqref{eq:4.1.17} were false for some positive", "$\\epsilon$, there would be an integer $K$ such that", "$$", "s_n\\le\\overline{s}-\\epsilon\\mbox{\\quad if\\quad} n\\ge K.", "$$", "However, this implies that", "$$", "M_k\\le\\overline{s}-\\epsilon\\mbox{\\quad if\\quad} k\\ge K,", "$$", "which contradicts \\eqref{eq:4.1.20}. Therefore, $\\overline{s}$", " has the stated properties.", "Now we must show that", "$\\overline{s}$ is the only real number with the stated properties.", "If $t<\\overline{s}$, the inequality", "$$", "s_n t-\\frac{t-\\overline{s}}{2}=\\overline{s}+\\frac{t-\\overline{s}}{", "2}", "$$", "cannot hold for infinitely many $n$, because this would contradict", "\\eqref{eq:4.1.16} with $\\epsilon=(t-\\overline{s})/2$. Therefore,", "$\\overline{s}$ is the only real number with the stated properties." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.1.6" ], "ref_ids": [ 83 ] } ], "ref_ids": [] }, { "id": 87, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.11", "categories": [], "title": "", "contents": [ "Every sequence $\\{s_n\\}$ of real numbers has a unique limit", "superior$,$", "$\\overline{s},$ and a unique limit inferior$,$ $\\underline{s}$, in the", "extended reals$,$ and", "\\begin{equation}\\label{eq:4.1.21}", "\\underline{s}\\le \\overline{s}.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "The existence and uniqueness of $\\overline{s}$ and", "$\\underline{s}$ follow from Theorem~\\ref{thmtype:4.1.9} and", "Definition~\\ref{thmtype:4.1.10}. If $\\overline{s}$ and $\\underline{s}$ are", "both finite, then \\eqref{eq:4.1.16} and \\eqref{eq:4.1.18} imply that", "$$", "\\underline{s}-\\epsilon<\\overline{s}+\\epsilon", "$$", "for every $\\epsilon>0$, which implies \\eqref{eq:4.1.21}. If", "$\\underline{s}=-\\infty$ or $\\overline{s}=\\infty$, then \\eqref{eq:4.1.21}", "is obvious. If $\\underline{s}=\\infty$ or $\\overline{s}=-\\infty$, then", "\\eqref{eq:4.1.21} follows immediately from Definition~\\ref{thmtype:4.1.10}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.1.9", "TRENCH_REAL_ANALYSIS-thmtype:4.1.10", "TRENCH_REAL_ANALYSIS-thmtype:4.1.10" ], "ref_ids": [ 86, 327, 327 ] } ], "ref_ids": [] }, { "id": 88, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.12", "categories": [], "title": "", "contents": [ "If $\\{s_n\\}$ is a sequence of real numbers, then", "\\begin{equation}\\label{eq:4.1.22}", "\\lim_{n\\to\\infty} s_n=s", "\\end{equation}", "if and only if", "\\begin{equation}\\label{eq:4.1.23}", "\\limsup_{n\\to\\infty}s_n=\\liminf_{n\\to\\infty} s_n=s.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "If $s=\\pm\\infty$, the equivalence of \\eqref{eq:4.1.22} and", "\\eqref{eq:4.1.23} follows immediately from their definitions. If", "$\\lim_{n\\to\\infty}s_n=s$ (finite), then Definition~\\ref{thmtype:4.1.1}", "implies that \\eqref{eq:4.1.16}--\\eqref{eq:4.1.19} hold with $\\overline{s}$ and $\\underline{s}$ replaced by", "$s$. Hence, \\eqref{eq:4.1.23} follows from the uniqueness of", "$\\overline{s}$ and $\\underline{s}$. For the converse, suppose that", "$\\overline{s}=\\underline{s}$ and let $s$ denote their common value.", "Then \\eqref{eq:4.1.16} and \\eqref{eq:4.1.18} imply that", "$$", "s-\\epsilon0,$ there is an integer $N$ such that", "\\begin{equation}\\label{eq:4.1.24}", "|s_n-s_m|<\\epsilon\\mbox{\\quad if\\quad} m,n\\ge N.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\lim_{n\\to\\infty}s_n=s$ and $\\epsilon>0$.", "By Definition~\\ref{thmtype:4.1.1}, there is an integer $N$ such that", "$$", "|s_r-s|<\\frac{\\epsilon}{2}\\mbox{\\quad if\\quad} r\\ge N.", "$$", "Therefore,", "$$", "|s_n-s_m|=|(s_n-s)+(s-s_m)|\\le |s_n-s|+|s-s_m|<\\epsilon", "\\mbox{\\quad if\\quad} n,m\\ge N.", "$$", "Therefore, the stated condition is necessary for convergence of", "$\\{s_n\\}$. To see that it is sufficient, we first observe that it", "implies that $\\{s_n\\}$ is bounded (Exercise~\\ref{exer:4.1.27}), so", "$\\overline{s}$ and $\\underline{s}$ are finite", "(Theorem~\\ref{thmtype:4.1.9}).", "Now suppose that $\\epsilon>0$ and $N$ satisfies \\eqref{eq:4.1.24}. From", "\\eqref{eq:4.1.16} and \\eqref{eq:4.1.17},", "\\begin{equation}\\label{eq:4.1.25}", "|s_n-\\overline{s}|<\\epsilon,", "\\end{equation}", "for some integer $n>N$ and, from \\eqref{eq:4.1.18} and \\eqref{eq:4.1.19},", "\\begin{equation}\\label{eq:4.1.26}", "|s_m-\\underline{s}|<\\epsilon", "\\end{equation}", "for some integer $m>N$. Since", "\\begin{eqnarray*}", "|\\overline{s}-\\underline{s}|\\ar=|(\\overline{s}-s_n)+", "(s_n-s_m)+(s_m-\\underline{s})|\\\\", "\\ar\\le |\\overline{s}-s_n|+|s_n-s_m|+|s_m-\\underline{s}|,", "\\end{eqnarray*}", "\\eqref{eq:4.1.24}--\\eqref{eq:4.1.26} imply that", "$$", "|\\overline{s}-\\underline{s}|<3\\epsilon.", "$$", "Since $\\epsilon$ is an arbitrary positive number, this implies that", "$\\overline{s}=\\underline{s}$, so $\\{s_n\\}$ converges, by", "Theorem~\\ref{thmtype:4.1.12}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.1.1", "TRENCH_REAL_ANALYSIS-thmtype:4.1.9", "TRENCH_REAL_ANALYSIS-thmtype:4.1.12" ], "ref_ids": [ 324, 86, 88 ] } ], "ref_ids": [] }, { "id": 90, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.2", "categories": [], "title": "", "contents": [ "If", "\\begin{equation}\\label{eq:4.2.1}", "\\lim_{n\\to\\infty}s_n=s\\quad (-\\infty\\le s\\le\\infty),", "\\end{equation}", "then", "\\begin{equation}\\label{eq:4.2.2}", "\\lim_{k\\to\\infty} s_{n_k}=s", "\\end{equation}", "for every subsequence $\\{s_{n_k}\\}$ of $\\{s_n\\}.$" ], "refs": [], "proofs": [ { "contents": [ "We consider the case where $s$ is finite and leave the rest to you", "(Exercise~\\ref{exer:4.2.4}). If \\eqref{eq:4.2.1} holds and $\\epsilon>0$,", "there is an integer $N$ such that", "$$", "|s_n-s|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N.", "$$", "Since $\\{n_k\\}$ is an increasing sequence, there is an integer $K$", "such that", "$n_k\\ge N$ if $k\\ge K$. Therefore,", "$$", "|s_{n_k}-L|<\\epsilon\\mbox{\\quad if\\quad} k\\ge K,", "$$", "which implies \\eqref{eq:4.2.2}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 91, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.3", "categories": [], "title": "", "contents": [ " If $\\{s_n\\}$ is monotonic and has a", "subsequence $\\{s_{n_k}\\}$ such that", "$$", "\\lim_{k\\to\\infty} s_{n_k}=s\\quad (-\\infty\\le s\\le\\infty),", "$$", "then", "$$", "\\lim_{n\\to\\infty} s_n=s.", "$$" ], "refs": [], "proofs": [ { "contents": [ "We consider the case where $\\{s_n\\}$ is nondecreasing and leave", "the rest to you (Exercise~\\ref{exer:4.2.6}). Since $\\{s_{n_k}\\}$ is also", "nondecreasing in this case, it suffices to show that", "\\begin{equation}\\label{eq:4.2.3}", "\\sup\\{s_{n_k}\\}=\\sup\\{s_n\\}", "\\end{equation}", "and then apply Theorem~\\ref{thmtype:4.1.6}\\part{a}. Since the", "set of terms of", "$\\{s_{n_k}\\}$ is contained in the set of terms of $\\{s_n\\}$,", "\\begin{equation} \\label{eq:4.2.4}", "\\sup\\{s_n\\}\\ge\\sup\\{s_{n_k}\\}.", "\\end{equation}", "Since $\\{s_n\\}$ is nondecreasing, there is for every $n$ an integer", "$n_k$ such that $s_n\\le s_{n_k}$. This implies that", "$$", "\\sup\\{s_n\\}\\le\\,\\sup\\{s_{n_k}\\}.", "$$", "This and \\eqref{eq:4.2.4} imply \\eqref{eq:4.2.3}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.1.6" ], "ref_ids": [ 83 ] } ], "ref_ids": [] }, { "id": 92, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.4", "categories": [], "title": "", "contents": [ "A point $\\overline{x}$ is a limit", "point of a set $S$ if and only if there is a sequence $\\{x_n\\}$ of points", "in $S$ such that $x_n\\ne\\overline{x}$ for $n\\ge 1,$ and", "$$", "\\lim_{n\\to\\infty}x_n=\\overline{x}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "For sufficiency, suppose that the stated condition holds.", "Then, for each $\\epsilon>0$, there is an integer $N$ such", "that $0<|x_n-x|<\\epsilon$ if $n\\ge N$. Therefore, every", "$\\epsilon$-neighborhood of $\\overline{x}$ contains infinitely many", "points of $S$. This means that $\\overline{x}$ is a limit point of $S$.", "For necessity, let $\\overline{x}$ be a limit point of $S$. Then,", "for every integer $n\\ge1$,", "the interval $(\\overline{x}-1/n,\\overline{x}+1/n)$", "contains", "a point $x_n\\ (\\ne\\overline{x})$ in $S$. Since", "$|x_m-\\overline{x}|\\le1/n$ if $m\\ge n$, $\\lim_{n\\to\\infty}x_n=", "\\overline{x}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 93, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.5", "categories": [], "title": "", "contents": [ "\\vspace*{3pt}", "\\begin{alist}", "\\item % (a)", " If $\\{x_n\\}$ is bounded$,$ then", "$\\{x_n\\}$ has a convergent subsequence$.$", "\\vspace*{3pt}", "\\item % (b)", " If $\\{x_n\\}$ is unbounded above$,$", " then $\\{x_n\\}$ has a subsequence $\\{x_{n_k}\\}$ such that", "$$", "\\lim_{k\\to\\infty} x_{n_k}=\\infty.", "$$", "\\vspace*{3pt}", "\\item % (c)", " If $\\{x_n\\}$ is unbounded", "below$,$ then $\\{x_n\\}$ has a subsequence $\\{x_{n_k}\\}$ such that", "$$", "\\lim_{k\\to\\infty} x_{n_k}=-\\infty.", "$$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "We prove \\part{a} and leave \\part{b} and \\part{c} to you", "(Exercise~\\ref{exer:4.2.7}). Let", "$S$ be the set of distinct numbers that occur as terms of $\\{x_n\\}$.", "(For example, if $\\{x_n\\}=\\{(-1)^n\\}$, $S=\\{1,-1\\}$; if", "$\\{x_n\\}=\\{1,\\frac{1}{2}, 1, \\frac{1}{3}, \\dots, 1, 1/n, \\dots\\}$,", "$S=\\{1,\\frac{1}{2}, \\dots, 1/n, \\dots\\}$.) If $S$ contains only finitely", "many points, then some $\\overline{x}$ in $S$ occurs infinitely often", "in $\\{x_n\\}$; that is, $\\{x_n\\}$ has a subsequence $\\{x_{n_k}\\}$ such", "that $x_{n_k}=\\overline{x}$ for all $k$. Then", "$\\lim_{k\\to\\infty}", "x_{n_k}=\\overline{x}$, and we are finished in this case.", "If $S$ is infinite, then, since $S$ is bounded (by assumption), the", "Bolzano--Weierstrass theorem (Theorem~\\ref{thmtype:1.3.8})", "implies that", "$S$ has a limit point", "$\\overline{x}$. From Theorem~\\ref{thmtype:4.2.4}, there is a sequence of", "points $\\{y_j\\}$ in $S$, distinct from $\\overline{x}$, such that", "\\begin{equation}\\label{eq:4.2.5}", "\\lim_{j\\to\\infty} y_j=\\overline{x}.", "\\end{equation}", "Although each $y_j$ occurs as a term of $\\{x_n\\}$, $\\{y_j\\}$ is", "not necessarily a subsequence of $\\{x_n\\}$, because if we write", "$$", "y_j=x_{n_j},", "$$", "there is no reason to expect that $\\{n_j\\}$ is an increasing sequence", "as required in Definition~\\ref{thmtype:4.2.1}. However, it is always", "possible to pick a subsequence $\\{n_{j_k}\\}$ of $\\{n_j\\}$ that is", "increasing, and then the sequence $\\{y_{j_k}\\}=\\{s_{n_{j_k}}\\}$ is a", "subsequence of both $\\{y_j\\}$ and $\\{x_n\\}$. Because of \\eqref{eq:4.2.5}", "and Theorem~\\ref{thmtype:4.2.2} this subsequence converges", "to~$\\overline{x}$.", "\\mbox{}" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:1.3.8", "TRENCH_REAL_ANALYSIS-thmtype:4.2.4", "TRENCH_REAL_ANALYSIS-thmtype:4.2.1", "TRENCH_REAL_ANALYSIS-thmtype:4.2.2" ], "ref_ids": [ 12, 92, 328, 90 ] } ], "ref_ids": [] }, { "id": 94, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.6", "categories": [], "title": "", "contents": [ "Let $f$ be defined on a closed interval $[a,b]$ containing", "$\\overline{x}.$ Then $f$ is continuous at $\\overline{x}$", "$($from the right if $\\overline{x}=a,$ from the left if", "$\\overline{x}=b$$)$ if and only if", "\\begin{equation}\\label{eq:4.2.6}", "\\lim_{n\\to\\infty} f(x_n)=f(\\overline{x})", "\\end{equation}", "whenever $\\{x_n\\}$ is a sequence of points in $[a,b]$ such that", "\\begin{equation}\\label{eq:4.2.7}", "\\lim_{n\\to\\infty} x_n=\\overline{x}.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Assume that $a<\\overline{x}0$, there is a", "$\\delta> 0$ such that", "\\begin{equation} \\label{eq:4.2.8}", "|f(x)-f(\\overline{x})|<\\epsilon\\mbox{\\quad if\\quad} |x-\\overline{x}|", "<\\delta.", "\\end{equation}", "From \\eqref{eq:4.2.7}, there is an integer $N$ such that", "$|x_n-\\overline{x}|<\\delta$", " if $n\\ge N$. This and \\eqref{eq:4.2.8} imply that", "$|f(x_n)-f(\\overline{x})|<\\epsilon$ if $n\\ge N$. This implies", "\\eqref{eq:4.2.6}, which shows that the stated condition is necessary.", "For sufficiency, suppose that $f$ is discontinuous at $\\overline{x}$.", "Then there is an $\\epsilon_0>0$ such that, for each positive integer", "$n$, there is a point $x_n$ that satisfies the inequality", "$$", "|x_n-\\overline{x}|<\\frac{1}{ n}", "$$", "\\newpage", "\\noindent", "while", "$$", "|f(x_n)-f(\\overline{x})|\\ge\\epsilon_0.", "$$", "The sequence $\\{x_n\\}$ therefore satisfies \\eqref{eq:4.2.7}, but not", "\\eqref{eq:4.2.6}. Hence, the stated condition cannot hold if $f$ is", "discontinuous at $\\overline{x}$. This proves sufficiency." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 95, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.7", "categories": [], "title": "", "contents": [ "If $f$ is continuous on a closed", "interval $[a,b],$ then $f$ is bounded on $[a,b].$" ], "refs": [], "proofs": [ { "contents": [ "The proof is by contradiction.", "If $f$ is not bounded on $[a,b]$, there is for each positive", "integer $n$ a point $x_n$ in $[a,b]$ such that", "$|f(x_n)|>n$. This implies that", "\\begin{equation}\\label{eq:4.2.9}", "\\lim_{n\\to\\infty}|f(x_n)|=\\infty.", "\\end{equation}", "Since $\\{x_n\\}$ is bounded, $\\{x_n\\}$ has a convergent subsequence", "$\\{x_{n_k}\\}$ (Theorem~\\ref{thmtype:4.2.5}\\part{a}). If", "$$", "\\overline{x}=\\lim_{k\\to\\infty} x_{n_k},", "$$", "then $\\overline{x}$ is a limit point of $[a,b]$, so", "$\\overline{x}\\in [a,b]$. If $f$ is continuous on $[a,b]$, then", "$$", "\\lim_{k\\to\\infty} f(x_{n_k})=f(\\overline{x})", "$$", "by Theorem~\\ref{thmtype:4.2.6}, so", "$$", "\\lim_{k\\to\\infty} |f(x_{n_k})|=|f(\\overline{x})|", "$$", "(Exercise~\\ref{exer:4.1.6}), which contradicts", "\\eqref{eq:4.2.9}.", "Therefore, $f$ cannot be both continuous and unbounded on $[a,b]$" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.2.5", "TRENCH_REAL_ANALYSIS-thmtype:4.2.6" ], "ref_ids": [ 93, 94 ] } ], "ref_ids": [] }, { "id": 96, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.2", "categories": [], "title": "", "contents": [ "The sum of a convergent series is unique$.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 97, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.3", "categories": [], "title": "", "contents": [ "Let", "$$", "\\sum_{n=k}^\\infty a_n=A\\mbox{\\quad and\\quad}\\sum_{n=k}^\\infty b_n=B,", "$$", "where $A$ and $B$ are finite$.$ Then", "$$", "\\sum_{n=k}^\\infty (ca_n)=cA", "$$", "if $c$ is a constant$,$", "$$", "\\sum_{n=k}^\\infty (a_n+b_n)=A+B,", "$$", "and", "$$", "\\sum_{n=k}^\\infty (a_n-b_n)=A-B.", "$$", "These relations also hold if one or both of $A$ and $B$ is infinite,", "provided that the right sides are not indeterminate$.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 98, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.5", "categories": [], "title": "Cauchy's Convergence Criterion for Series", "contents": [ "A series $\\sum a_n$ converges if and only if for every", "$\\epsilon>0$", "there is an integer $N$ such that", "\\begin{equation}\\label{eq:4.3.3}", "|a_n+a_{n+1}+\\cdots+a_m|<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "In terms of the partial sums $\\{A_n\\}$ of $\\sum a_n$,", "$$", "a_n+a_{n+1}+\\cdots+a_m=A_m-A_{n-1}.", "$$", "Therefore, \\eqref{eq:4.3.3} can be written as", "$$", "|A_m-A_{n-1}|<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N.", "$$", "Since $\\sum a_n$ converges if and only if $\\{A_n\\}$ converges,", "Theorem~\\ref{thmtype:4.1.13} implies the conclusion." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.1.13" ], "ref_ids": [ 89 ] } ], "ref_ids": [] }, { "id": 99, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.8", "categories": [], "title": "", "contents": [ "If $a_n\\ge0$ for $n\\ge k,$ then $\\sum a_n$ converges if its partial", "sums are bounded$,$ or diverges to $\\infty$ if they are not$.$ These", "are the only possibilities and$,$ in either case$,$", "$$", "\\sum_{n=k}^\\infty a_n =\\,\\sup\\set{A_n}{n\\ge k}\\negthickspace,", "$$", "where", "$$", "A_n=a_k+a_{k+1}+\\cdots+a_n,\\quad n\\ge k.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $A_n=A_{n-1}+a_n$ and $a_n\\ge0$ $(n\\ge k)$, the sequence", "$\\{A_n\\}$ is nondecreasing, so the conclusion follows from", "Theorem~\\ref{thmtype:4.1.6}\\part{a} and", "Definition~\\ref{thmtype:4.3.1}.", "\\newline\\mbox{}" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.1.6", "TRENCH_REAL_ANALYSIS-thmtype:4.3.1" ], "ref_ids": [ 83, 329 ] } ], "ref_ids": [] }, { "id": 100, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.9", "categories": [], "title": "The Comparison Test", "contents": [ "Suppose that", "\\begin{equation}\\label{eq:4.3.5}", "0\\le a_n\\le b_n,\\quad n\\ge k.", "\\end{equation}", "Then", "\\begin{alist}", "\\item % (a)", " $\\sum a_n<\\infty$ if $\\sum b_n<\\infty$$.$", "\\item % (b)", " $\\sum b_n=\\infty$ if $\\sum a_n=\\infty.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "\\part{a} If", "$$", "A_n=a_k+a_{k+1}+\\cdots+a_n\\mbox{\\quad and\\quad} B_n=b_k+", "b_{k+1}+\\cdots+b_n,\\quad n\\ge k,", "$$", "then, from \\eqref{eq:4.3.5},", "\\begin{equation}\\label{eq:4.3.6}", "A_n\\le B_n.", "\\end{equation}", "Now we use Theorem~\\ref{thmtype:4.3.8}.", "If $\\sum b_n<\\infty$, then $\\{B_n\\}$ is bounded above", " and \\eqref{eq:4.3.6} implies that $\\{A_n\\}$ is", "also; therefore, $\\sum a_n<\\infty$.", "On the other hand, if", " $\\sum a_n=\\infty$, then $\\{A_n\\}$ is unbounded above", " and \\eqref{eq:4.3.6} implies that $\\{B_n\\}$ is", "also; therefore, $\\sum b_n~=~\\infty$.", "\\vspace*{4pt}", "We leave it to you to show that \\part{a} implies \\part{b}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.3.8" ], "ref_ids": [ 99 ] } ], "ref_ids": [] }, { "id": 101, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.10", "categories": [], "title": "The Integral Test", "contents": [ "Let", "\\begin{equation}\\label{eq:4.3.7}", "c_n=f(n),\\quad n\\ge k,", "\\end{equation}", "where $f$ is positive$,$ nonincreasing$,$ and locally integrable on", "$[k,\\infty).$", "Then", "\\begin{equation}\\label{eq:4.3.8}", "\\sum c_n<\\infty", "\\end{equation}", "if and only if", "\\begin{equation}\\label{eq:4.3.9}", "\\int^\\infty_k f(x)\\,dx<\\infty.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "We first observe that \\eqref{eq:4.3.9} holds if and only if", "\\begin{equation}\\label{eq:4.3.10}", "\\sum_{n=k}^\\infty \\int^{n+1}_n f(x)\\,dx<\\infty", "\\end{equation}", "(Exercise~\\ref{exer:4.3.9}), so it is enough to show that \\eqref{eq:4.3.8}", "holds if and only if \\eqref{eq:4.3.10} does. From \\eqref{eq:4.3.7} and the", "assumption that $f$ is nonincreasing,", "$$", "c_{n+1}=f(n+1)\\le f(x)\\le f(n)=c_n,\\quad n\\le x\\le n+1,\\quad n\\ge k.", "$$", "Therefore,", "$$", "c_{n+1}=\\int^{n+1}_n c_{n+1}\\,dx\\le\\int^{n+1}_n f(x)\\,dx\\le", "\\int^{n+1}_n c_n\\,dx=c_n,\\quad n\\ge k", "$$", "(Theorem~\\ref{thmtype:3.3.4}). From the first inequality and", "Theorem~\\ref{thmtype:4.3.9}\\part{a} with $a_n=c_{n+1}$ and", "$b_n=\\int^{n+1}_n", "f(x)\\,dx$, \\eqref{eq:4.3.10} implies that $\\sum c_{n+1}<\\infty$, which is", "equivalent to \\eqref{eq:4.3.8}. From the second inequality and", "Theorem~\\ref{thmtype:4.3.9}\\part{a} with $a_n=\\int^{n+1}_n f(x)\\,dx$ and", "$b_n=c_n$, \\eqref{eq:4.3.8} implies \\eqref{eq:4.3.10}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.3.4", "TRENCH_REAL_ANALYSIS-thmtype:4.3.9", "TRENCH_REAL_ANALYSIS-thmtype:4.3.9" ], "ref_ids": [ 56, 100, 100 ] } ], "ref_ids": [] }, { "id": 102, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.11", "categories": [], "title": "", "contents": [ "Suppose that $a_n\\ge0$ and $b_n>0$ for $n\\ge k.$ Then", "\\begin{alist}", "\\item % (a)", "$\\dst{\\sum a_n<\\infty\\mbox{\\quad if\\quad}\\sum b_n<", "\\infty\\mbox{\\quad and\\quad}\\limsup_{n\\to\\infty} a_n/b_n<\\infty}.$", "\\item % (b)", " $\\dst{\\sum a_n=\\infty\\mbox{\\quad if\\quad}\\sum b_n=", "\\infty\\mbox{\\quad and\\quad}\\liminf_{n\\to\\infty} a_n/b_n>0}.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "\\part{a} If", "$\\limsup_{n\\to\\infty} a_n/b_n<\\infty$, then $\\{a_n/b_n\\}$ is", "bounded, so there is a constant $M$ and an integer $k$ such that", "$$", "a_n\\le Mb_n,\\quad n\\ge k.", "$$", "Since $\\sum b_n<\\infty$, Theorem~\\ref{thmtype:4.3.3} implies that $\\sum", "(Mb_n)< \\infty$. Now", "$\\sum a_n<\\infty$, by the comparison test.", "\\part{b}", "If", "$\\liminf_{n\\to\\infty} a_n/b_n>0$,", " there is a constant $m$ and an integer $k$ such that", "$$", "a_n\\ge mb_n,\\quad n\\ge k.", "$$", "Since $\\sum b_n=\\infty$, Theorem~\\ref{thmtype:4.3.3} implies that $\\sum", "(mb_n)= \\infty$. Now", "$\\sum a_n=\\infty$, by the comparison test." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.3.3", "TRENCH_REAL_ANALYSIS-thmtype:4.3.3" ], "ref_ids": [ 97, 97 ] } ], "ref_ids": [] }, { "id": 103, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.13", "categories": [], "title": "", "contents": [ "Suppose that $a_n>0,$ $b_n>0,$ and", "\\begin{equation}\\label{eq:4.3.12}", "\\frac{a_{n+1}}{ a_n}\\le \\frac{b_{n+1}}{ b_n}.", "\\end{equation}", "Then", "\\begin{alist}", "\\item % (a)", " $\\sum a_n<\\infty$ if $\\sum b_n<\\infty.$", "\\item % (b)", " $\\sum b_n=\\infty$ if $\\sum a_n=\\infty.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "Rewriting \\eqref{eq:4.3.12} as", "$$", "\\frac{a_{n+1}}{ b_{n+1}}\\le \\frac{a_n}{ b_n},", "$$", "we see that $\\{a_n/b_n\\}$ is nonincreasing. Therefore,", "$\\limsup_{n \\to\\infty} a_n/b_n<\\infty$, and", "Theorem~\\ref{thmtype:4.3.11}\\part{a} implies \\part{a}.", "To prove", "\\part{b}, suppose that $\\sum a_n=\\infty$. Since $\\{a_n/b_n\\}$", "is nonincreasing,", " there is a number $\\rho$", "such that $b_n\\ge \\rho a_n$ for large $n$. Since $\\sum (\\rho", "a_n)=\\infty$ if $\\sum a_n=\\infty$, Theorem~\\ref{thmtype:4.3.9}\\part{b}", "(with $a_n$ replaced by $\\rho a_n$)", "implies that $\\sum b_n=\\infty$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.3.11", "TRENCH_REAL_ANALYSIS-thmtype:4.3.9" ], "ref_ids": [ 102, 100 ] } ], "ref_ids": [] }, { "id": 104, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.14", "categories": [], "title": "The Ratio Test", "contents": [ "Suppose that $a_n>0$ for $n\\ge k.$ Then", "\\vspace*{5pt}", "\\begin{alist}", "\\vspace*{5pt}", "\\item % (a)", "$\\sum a_n<\\infty$ if\\,", "$\\limsup_{n\\to\\infty} a_{n+1}/a_n<1.$", "\\vspace*{5pt}", "\\item % (b)", " $\\sum a_n=\\infty$ if\\,", "$\\liminf_{n\\to\\infty} a_{n+1}/a_n>1.$", "\\end{alist}", "\\vspace*{5pt}", "\\noindent If", "\\begin{equation}\\label{eq:4.3.13}", "\\liminf_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}\\le1\\le", "\\limsup_{n\\to\\infty}\\frac{a_{n+1}}{ a_n},", "\\end{equation}", "then the test is inconclusive$;$ that is$,$ $\\sum a_n$ may converge", "or diverge$.$" ], "refs": [], "proofs": [ { "contents": [ "\\part{a} If", "$$", "\\limsup_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}<1,", "$$", "there is a number $r$ such that $01,", "$$", " there is a number $r$ such that $r>1$ and", "$$", "\\frac{a_{n+1}}{ a_n}>r", "$$", "for $n$ sufficiently large. This can be rewritten as", "$$", "\\frac{a_{n+1}}{ a_n}>\\frac{r^{n+1}}{ r^n}.", "$$", "Since $\\sum r^n=\\infty$,", "Theorem~\\ref{thmtype:4.3.13}\\part{b} with $a_n=r^n$ implies that $\\sum", "b_n=\\infty$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.3.13", "TRENCH_REAL_ANALYSIS-thmtype:4.3.13" ], "ref_ids": [ 103, 103 ] } ], "ref_ids": [] }, { "id": 105, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.16", "categories": [], "title": "", "contents": [ "Suppose that $a_n>0$ for large $n.$ Let", "$$", "M=\\limsup_{n\\to\\infty} n\\left(\\frac{a_{n+1}}{ a_n}-", "1\\right)\\mbox{\\quad and\\quad} m=\\liminf_{n\\to\\infty} n", "\\left(\\frac{a_{n+1}}{ a_n}-1\\right).", "$$", "Then", "\\begin{alist}", "\\item % (a)", " $\\sum a_n<\\infty$ if $M<-1.$", "\\item % (b)", " $\\sum a_n=\\infty$ if $m>-1.$", "\\end{alist}", "The test is inconclusive if $m\\le-1\\le M.$" ], "refs": [], "proofs": [ { "contents": [ "\\part{a}", "We need the inequality", "\\begin{equation}\\label{eq:4.3.15}", "\\frac{1}{(1+x)^p}>1-px,\\quad x>0,\\ p>0.", "\\end{equation}", "This follows from Taylor's theorem", "(Theorem~\\ref{thmtype:2.5.4}), which implies that", "$$", "\\frac{1}{(1+x)^p}=1-px+\\frac{1}{2}\\frac{p(p+1)}{(1+c)^{p+2}}x^2,", "$$", "where $00$,", "this implies \\eqref{eq:4.3.15}.", "Now suppose that $M<-p<-1$. Then there is an integer $k$ such that", "$$", "n\\left(\\frac{a_{n+1}}{ a_n}-1\\right)<-p,\\quad n\\ge k,", "$$", "so", "$$", "\\frac{a_{n+1}}{ a_n}<1-\\frac{p}{ n},\\quad n\\ge k.", "$$", "Hence,", "$$", "\\frac{a_{n+1}}{ a_n}<\\frac{1}{(1+1/n)^p},\\quad n\\ge k,", "$$", "as can be seen by letting $x=1/n$ in \\eqref{eq:4.3.15}. From this,", "$$", "\\frac{a_{n+1}}{ a_n}<\\frac{1}{(n+1)^p}\\bigg/\\frac{1}{ n^p},\\quad n\\ge k.", "$$", " Since $\\sum 1/n^p<\\infty$ if $p>1$,", " Theorem~\\ref{thmtype:4.3.13}\\part{a} implies that", " $\\sum a_n<\\infty$.", "\\part{b} Here we need the inequality", "\\begin{equation}\\label{eq:4.3.16}", "(1-x)^q<1-qx,\\quad 0-q,\\quad n\\ge k,", "$$", "so", "$$", "\\frac{a_{n+1}}{ a_n}\\ge1-\\frac{q}{ n},\\quad n\\ge k.", "$$", "If $q\\le0$, then $\\sum a_n=\\infty$, by Corollary~\\ref{thmtype:4.3.6}.", "Hence, we may assume that $0\\left(1-\\frac{1}{ n}\\right)^q,\\quad n\\ge k,", "$$", "\\newpage", "\\noindent", "as can be seen by setting $x=1/n$ in \\eqref{eq:4.3.16}. Hence,", "$$", "\\frac{a_{n+1}}{ a_n}>\\frac{1}{ n^q}\\bigg/\\frac{1}{(n-1)^q},\\quad n\\ge k.", "$$", " Since $\\sum 1/n^q=\\infty$ if $q<1$,", " Theorem~\\ref{thmtype:4.3.13}\\part{b} implies that", " $\\sum a_n=\\infty$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.5.4", "TRENCH_REAL_ANALYSIS-thmtype:4.3.13", "TRENCH_REAL_ANALYSIS-thmtype:4.3.6", "TRENCH_REAL_ANALYSIS-thmtype:4.3.13" ], "ref_ids": [ 42, 103, 277, 103 ] } ], "ref_ids": [] }, { "id": 106, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.17", "categories": [], "title": "Cauchy's Root Test", "contents": [ "If $a_n\\ge 0$ for $n\\ge k,$ then", "\\begin{alist}", "\\item % (a)", " $\\sum a_n<\\infty$ if", "$\\limsup_{n\\to\\infty} a^{1/n}_n<1.$", "\\item % (b)", " $\\sum a_n=\\infty$ if", "$\\limsup_{n\\to\\infty} a^{1/n}_n>1.$", "\\end{alist}", "The test is inconclusive if $\\limsup_{n\\to\\infty} a^{1/n}_n=", "1.$" ], "refs": [], "proofs": [ { "contents": [ "\\part{a} If $\\limsup_{n\\to\\infty}a^{1/n}_n<1$, there is an", " $r$", "such that $01$, then $a^{1/n}_n>1$", "for infinitely many values of $n$,", "so $\\sum a_n=\\infty$, by", "Corollary~\\ref{thmtype:4.3.6}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.3.6" ], "ref_ids": [ 277 ] } ], "ref_ids": [] }, { "id": 107, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.19", "categories": [], "title": "", "contents": [ "absolutely$,$ then $\\sum a_n$ converges$.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 108, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.20", "categories": [], "title": "Dirichlet's Test for Series", "contents": [ "The series $\\sum ^\\infty_{n=k} a_nb_n$ converges if $\\lim_{n\\to\\infty}", "a_n= 0,$", "\\begin{equation}\\label{eq:4.3.18}", "\\sum |a_{n+1}-a_n|<\\infty,", "\\end{equation}", "and", "\\begin{equation}\\label{eq:4.3.19}", "|b_k+b_{k+1}+\\cdots+b_n|\\le M,\\quad n\\ge k,", "\\end{equation}", "for some constant $M.$" ], "refs": [], "proofs": [ { "contents": [ "The proof is similar to the proof of Dirichlet's test for integrals.", "Define", "$$", "B_n=b_k+b_{k+1}+\\cdots+b_n,\\quad n\\ge k", "$$", "and consider the partial sums of $\\sum_{n=k}^\\infty a_nb_n$:", "\\begin{equation}\\label{eq:4.3.20}", "S_n=a_kb_k+a_{k+1}b_{k+1}+\\cdots+a_nb_n,\\quad n\\ge k.", "\\end{equation}", "By substituting", "$$", "b_k=B_k\\mbox{\\quad and\\quad} b_n=B_n-B_{n-1},\\quad n\\ge k+1,", "$$", "into \\eqref{eq:4.3.20}, we obtain", "$$", "S_n=a_kB_k+a_{k+1}(B_{k+1}-B_k)+\\cdots+a_n(B_n-B_{n-1}),", "$$", "which we rewrite as", "\\begin{equation}\\label{eq:4.3.21}", "\\begin{array}{rcl}", "S_n\\ar=(a_k-a_{k+1})B_k+(a_{k+1}-a_{k+2})B_{k+1}+\\cdots\\\\", "\\ar{}+\\,(a_{n-1}-a_n)B_{n-1}+a_nB_n.", "\\end{array}", "\\end{equation}", "\\newpage", "\\noindent", "(The procedure that led from \\eqref{eq:4.3.20} to \\eqref{eq:4.3.21} is called", "{\\it summation by parts\\/}. It is analogous", "to integration by parts.) Now \\eqref{eq:4.3.21} can be viewed as", "\\begin{equation}\\label{eq:4.3.22}", "S_n=T_{n-1}+a_nB_n,", "\\end{equation}", "where", "$$", "T_{n-1}=(a_k-a_{k+1})B_k+(a_{k+1}-a_{k+2})", "B_{k+1}+\\cdots+(a_{n-1}-a_n)B_{n-1};", "$$", "that is, $\\{T_n\\}$ is the sequence of partial sums of the series", "\\begin{equation}\\label{eq:4.3.23}", "\\sum_{j=k}^\\infty (a_j-a_{j+1})B_j.", "\\end{equation}", "Since", "$$", "|(a_j-a_{j+1})B_j|\\le M|a_j-a_{j+1}|", "$$", "from \\eqref{eq:4.3.19}, the comparison test and \\eqref{eq:4.3.18} imply that", "the series \\eqref{eq:4.3.23} converges absolutely.", "Theorem~\\ref{thmtype:4.3.19}", "now implies that $\\{T_n\\}$ converges. Let $T=\\lim_{n\\to\\infty}T_n$.", "Since $\\{B_n\\}$ is bounded and $\\lim_{n\\to \\infty}a_n=0$, we infer", "from \\eqref{eq:4.3.22} that", "$$", "\\lim_{n\\to\\infty} S_n=\\lim_{n\\to\\infty}T_{n-1}+\\lim_{n\\to", "\\infty}a_nB_n=T+0=T.", "$$", "Therefore, $\\sum a_nb_n$ converges." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.3.19" ], "ref_ids": [ 107 ] } ], "ref_ids": [] }, { "id": 109, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.23", "categories": [], "title": "", "contents": [ "Suppose that $\\sum_{n=k}^\\infty a_n=A,$ where $-\\infty \\le A\\le\\infty.$ Let", "$\\{n_j\\}_1^\\infty$ be an increasing sequence of integers, with $n_1\\ge", "k$. Define", "\\begin{eqnarray*}", "b_1\\ar=a_k+\\cdots+a_{n_1},\\\\", "b_2\\ar=a_{{n_1}+1}+\\cdots+a_{n_2},\\\\", "&\\vdots\\\\", "b_r\\ar=a_{n_{r-1}+1}+\\cdots+a_{n_r}.", "\\end{eqnarray*}", "Then", "$$", "\\sum_{j=1}^\\infty b_{n_j}=A.", "$$" ], "refs": [], "proofs": [ { "contents": [ "If $T_r$ is the $r$th partial sum of $\\sum_{j=1}^\\infty", "b_{n_j}$ and $\\{A_n\\}$ is the $n$th partial sum of", "$\\sum_{s=k}^\\infty a_s$, then", "\\begin{eqnarray*}", "T_r\\ar=b_1+b_2+\\cdots+b_r\\\\", "\\ar=(a_1+\\cdots+a_{n_1})+(a_{n_1+1}+\\cdots+a_{n_2})+\\cdots+", "(a_{n_{r-1}+1}+\\cdots+a_{n_r})\\\\", "\\ar=A_{n_r}.", "\\end{eqnarray*}", "Thus, $\\{T_r\\}$ is a subsequence of $\\{A_n\\}$, so", "$\\lim_{r\\to\\infty} T_r=\\lim_{n\\to\\infty}A_n=A$ by", "Theorem~\\ref{thmtype:4.2.2}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.2.2" ], "ref_ids": [ 90 ] } ], "ref_ids": [] }, { "id": 110, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.24", "categories": [], "title": "", "contents": [ "If $\\sum_{n=1}^\\infty b_n$ is a rearrangement of an absolutely", "convergent series $\\sum_{n=1}^\\infty a_n,$ then $\\sum_{n=1}^\\infty", "b_n$ also converges absolutely$,$ and to the same sum$.$" ], "refs": [], "proofs": [ { "contents": [ "Let", "$$", "\\overline{A}_n=|a_1|+|a_2|+\\cdots+|a_n|\\mbox{\\quad and\\quad}", "\\overline{B}_n=|b_1|+|b_2|+\\cdots+|b_n|.", "$$", "For each $n\\ge1$, there is an integer $k_n$ such that", "$b_1$, $b_2$, \\dots, $b_n$ are included among", "$a_1$, $a_2$, \\dots, $a_{k_n}$,", "so $\\overline{B}_n\\le\\overline{A}_{k_n}$. Since", "$\\{\\overline{A}_n\\}$ is bounded, so is $\\{\\overline{B}_n\\}$, and", "therefore $\\sum |b_n|<\\infty$ (Theorem~\\ref{thmtype:4.3.8}).", "Now let", "\\begin{eqnarray*}", "A_n\\ar=a_1+a_2+\\cdots+a_n,\\quad B_n=b_1+b_2+\\cdots+", "b_n,\\\\", "A\\ar=\\sum_{n=1}^\\infty a_n,\\mbox{\\quad and\\quad} B=\\sum_{n=1}^\\infty", "b_n.", "\\end{eqnarray*}", "\\newpage", "\\noindent", "We must show that $A=B$. Suppose that $\\epsilon>0$. From Cauchy's", "convergence criterion for series and the", "absolute convergence of $\\sum a_n$, there is an", "integer $N$ such that", "\\vspace*{2pt}", "$$", "|a_{N+1}|+|a_{N+2}|+\\cdots+|a_{N+k}|<\\epsilon,\\quad k\\ge1.", "$$", "\\vspace*{2pt}", "\\noindent\\hskip-.3em Choose $N_1$ so that $a_1$, $a_2$, \\dots, $a_N$", "are included", "among", "$b_1$, $b_2$, \\dots, $b_{N_1}$. If $n\\ge N_1$, then $A_n$ and $B_n$", "both", "include the terms $a_1$, $a_2$, \\dots, $a_N$, which cancel on", "subtraction;", "thus, $|A_n-B_n|$ is dominated by the sum of the absolute values of", "finitely many terms from $\\sum a_n$ with subscripts greater than $N$.", "Since every such sum is less than~$\\epsilon$,", "\\vspace*{2pt}", "$$", "|A_n-B_n|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N_1.", "$$", "\\vspace*{2pt}", "Therefore, $\\lim_{n\\to\\infty}(A_n-B_n)=0$ and $A=B$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.3.8" ], "ref_ids": [ 99 ] } ], "ref_ids": [] }, { "id": 111, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.25", "categories": [], "title": "", "contents": [ "If $P=\\{a_{n_i}\\}_1^\\infty$ and", "$Q=", "\\{a_{m_j}\\}_1^\\infty$ are respectively the subsequences of all", "positive and", "negative terms in a conditionally convergent series $\\sum a_n,$ then", "\\begin{equation} \\label{eq:4.3.24}", "\\sum_{i=1}^\\infty a_{n_i}=\\infty\\mbox{\\quad and\\quad}\\sum_{j=1}^\\infty", "a_{m_j}=-\\infty.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "If both series in \\eqref{eq:4.3.24} converge, then $\\sum", "a_n$ converges absolutely, while if one converges and the other", "diverges, then $\\sum a_n$ diverges to $\\infty$ or $-\\infty$. Hence,", "both must diverge." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 112, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.26", "categories": [], "title": "", "contents": [ "Suppose that $\\sum_{n=1}^\\infty a_n$ is conditionally convergent and", " $\\mu$ and $\\nu$ are arbitrarily given in the extended", "reals$,$ with $\\mu\\le\\nu.$ Then", "the terms of $\\sum_{n=1}^\\infty a_n$", "can be rearranged to form a series $\\sum_{n=1}^\\infty b_n$", "with partial sums", "$$", "B_n=b_1+b_2+\\cdots+b_n,\\quad n\\ge1,", "$$", "such that", "\\begin{equation}\\label{eq:4.3.25}", "\\limsup_{n\\to\\infty}B_n=\\nu\\mbox{\\quad and\\quad}", "\\liminf_{n\\to\\infty}B_n=\\mu.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "We consider the case where $\\mu$ and $\\nu$ are finite and leave", "the other cases to you (Exercise~\\ref{exer:4.3.36}).", "We may ignore any zero terms that occur in $\\sum_{n=1}^\\infty a_n$.", "For convenience, we", "denote the positive terms by", " $P=\\{\\alpha_i\\}_1^\\infty$ and and the negative terms by", "$Q=\\{-\\beta_j\\}_1^\\infty$. We construct the sequence", "\\begin{equation} \\label{eq:4.3.26}", "\\{b_n\\}_1^\\infty=\\{\\alpha_1, \\dots,\\alpha_{m_1},-\\beta_1, \\dots,-\\beta_{n_1},", "\\alpha_{m_1+1}, \\dots,\\alpha_{m_2},-\\beta_{n_1+1}, \\dots,-\\beta_{n_2},", "\\dots\\},", "\\end{equation}", "\\newpage", "\\noindent", "with segments chosen alternately from $P$ and $Q$. Let $m_0=n_0=0$.", "If $k\\ge1$, let $m_k$ and $n_k$ be the smallest integers such that", "$m_k>m_{k-1}$, $n_k>n_{k-1}$,", "$$", "\\sum_{i=1}^{m_k}\\alpha_i-\\sum_{j=1}^{n_{k-1}}\\beta_j\\ge\\nu,", "\\mbox{\\quad and \\quad}", "\\sum_{i=1}^{m_k}\\alpha_i-\\sum_{j=1}^{n_k}\\beta_j\\le\\mu.", "$$", "Theorem~\\ref{thmtype:4.3.25} implies", "that this construction is possible:", "since $\\sum \\alpha_i=\\sum\\beta_j=\\infty$, we", "can choose $m_k$ and $n_k$ so that", "$$", "\\sum_{i=m_{k-1}}^{m_k}\\alpha_i\\mbox{\\quad and\\quad}", "\\sum_{j=n_{k-1}}^{n_k}\\beta_j", "$$", "are as large as we please, no matter how large $m_{k-1}$ and $n_{k-1}$", "are (Exercise~\\ref{exer:4.3.23}).", "Since $m_k$ and $n_k$ are the smallest integers with the specified", "properties,", "\\begin{eqnarray}", "\\nu\\le B_{m_k+n_{k-1}}\\ar<\\nu+\\alpha_{m_k},\\quad k\\ge2,", "\\label{eq:4.3.27}\\\\", "\\arraytext{and}\\nonumber\\\\", "\\mu-\\beta_{n_k}\\ar0$ if $m_k+n_k< n\\le m_{k+1}+n_k$, so", "\\begin{equation}\\label{eq:4.3.30}", "B_{m_k+n_k}\\le B_n\\le B_{m_{k+1}+n_k},\\quad m_k+n_k\\le n\\le m_{k+1}+n_k.", "\\end{equation}", "Because of \\eqref{eq:4.3.27} and \\eqref{eq:4.3.28}, \\eqref{eq:4.3.29}", "and \\eqref{eq:4.3.30} imply that", "\\begin{eqnarray}", "\\mu-\\beta_{n_k}\\ar0$ then", " $B_n>\\nu+ \\epsilon$ for only finitely many", "values of $n$. Therefore,", "$\\limsup_{n\\to\\infty} B_n=\\nu$.", "From the second inequality in \\eqref{eq:4.3.28}, $B_n\\le \\mu$ for", "infinitely many values of $n$. However, since", "$\\lim_{j\\to\\infty}\\beta_j=0$,", "the first inequalities in \\eqref{eq:4.3.31} and \\eqref{eq:4.3.32}", "imply that if $\\epsilon>0$ then", " $B_n<\\mu-\\epsilon$ for only finitely many", "values of $n$. Therefore,", "$\\liminf_{n\\to\\infty} B_n=\\mu$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.3.25" ], "ref_ids": [ 111 ] } ], "ref_ids": [] }, { "id": 113, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.27", "categories": [], "title": "", "contents": [ "Let", "$$", "\\sum_{n=0}^\\infty a_n=A\\mbox{\\quad and\\quad}\\sum_{n=0}^\\infty b_n=B,", "$$", "where $A$ and $B$ are finite, and at least one term of each series", "is nonzero. Then $\\sum_{n=0}^\\infty p_n=AB$ for every sequence", "$\\{p_n\\}$ obtained by ordering the products in $\\eqref{eq:4.3.33}$ if and", "only if $\\sum a_n$ and $\\sum b_n$ converge absolutely$.$ Moreover$,$", "in this case, $\\sum p_n$ converges absolutely$.$" ], "refs": [], "proofs": [ { "contents": [ "First, let $\\{p_n\\}$ be the sequence obtained by", "arranging the products $\\{a_ib_j\\}$ according to the scheme indicated in", "\\eqref{eq:4.3.34}, and define", "$$", "\\begin{array}{ll}", "A_n=a_0+a_1+\\cdots+a_n,&", "\\overline{A}_n=|a_0|+|a_1|+\\cdots+|a_n|,\\\\[2\\jot]", "B_n=b_0+b_1+\\cdots+b_n,&", "\\overline{B}_n=|b_0|+|b_1|+\\cdots+|b_n|,\\\\[2\\jot]", "P_n\\hskip.1em=p_0+p_1+\\cdots+p_n,&\\overline{P}_n\\hskip.1em=|p_0|+|p_1|+\\cdots+|p_n|.", "\\end{array}", "$$", "From \\eqref{eq:4.3.34}, we see that", "$$", "P_0=A_0B_0,\\quad P_3=A_1B_1,\\quad P_8=A_2B_2,", "$$", "and, in general,", "\\begin{equation}\\label{eq:4.3.36}", "P_{(m+1)^2-1}=A_mB_m.", "\\end{equation}", "\\newpage", "\\noindent", "Similarly,", "\\begin{equation}\\label{eq:4.3.37}", "\\overline{P}_{(m+1)^2-1}=\\overline{A}_m\\overline{B}_m.", "\\end{equation}", "If $\\sum |a_n|<\\infty$ and $\\sum |b_n|<\\infty$, then", "$\\{\\overline{A}_m\\overline{B}_m\\}$ is bounded and, since", "$\\overline{P}_m\\le\\overline{P}_{(m+1)^2-1}$,", "\\eqref{eq:4.3.37} implies that $\\{\\overline{P}_m\\}$ is bounded. Therefore,", "$\\sum |p_n| <\\infty$, so $\\sum p_n$ converges. Now", "$$", "\\begin{array}{rcll}", "\\dst{\\sum ^\\infty_{n=0}p_n}\\ar=\\dst{\\lim_{n\\to\\infty}P_n}&\\mbox{(by", "definition)}\\\\[2\\jot]", "\\ar=\\dst{\\lim_{m\\to\\infty} P_{(m+1)^2-1}}&\\mbox{(by", "Theorem~\\ref{thmtype:4.2.2})}\\\\[2\\jot]", "\\ar=\\dst{\\lim_{m\\to\\infty} A_mB_m}&\\mbox{(from \\eqref{eq:4.3.36})}\\\\[2\\jot]", "\\ar=\\dst{\\left(\\lim_{m\\to\\infty}", "A_m\\right)\\left(\\lim_{m\\to\\infty}B_m\\right)}", "&\\mbox{(by Theorem~\\ref{thmtype:4.1.8})}\\\\[2\\jot]", "\\ar=AB.", "\\end{array}", "$$", "Since any other ordering of the products in \\eqref{eq:4.3.33} produces a", " a rearrangement of the", "absolutely convergent series $\\sum_{n=0}^\\infty p_n$,", "Theorem~\\ref{thmtype:4.3.24} implies that $\\sum |q_n|<\\infty$ for every", "such ordering and that $\\sum_{n=0}^\\infty q_n=AB$. This shows that", "the stated condition is sufficient.", "For necessity, again let $\\sum_{n=0}^\\infty p_n$ be obtained from the", "ordering indicated in \\eqref{eq:4.3.34}, and suppose that $\\sum_{n=0}^\\infty p_n$ and all its", "rearrangements converge to $AB$. Then $\\sum p_n$ must converge", "absolutely, by Theorem~\\ref{thmtype:4.3.26}. Therefore,", "$\\{\\overline{P}_{m^2-1}\\}$ is bounded, and \\eqref{eq:4.3.37} implies that", "$\\{\\overline{A}_m\\}$ and $\\{\\overline{B}_m\\}$ are bounded.", "(Here we need", "the assumption that neither $\\sum a_n$ nor $\\sum b_n$ consists", "entirely of zeros. Why?)", " Therefore,", "$\\sum |a_n|<\\infty$ and $\\sum |b_n|<\\infty$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.2.2", "TRENCH_REAL_ANALYSIS-thmtype:4.1.8", "TRENCH_REAL_ANALYSIS-thmtype:4.3.24", "TRENCH_REAL_ANALYSIS-thmtype:4.3.26" ], "ref_ids": [ 90, 85, 110, 112 ] } ], "ref_ids": [] }, { "id": 114, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.29", "categories": [], "title": "", "contents": [ "If $\\sum_{n=0}^\\infty a_n$ and", "$\\sum_{n=0}^\\infty b_n$ converge absolutely to sums $A$ and $B,$ then", "the Cauchy product of $\\sum_{n=0}^\\infty a_n$", "and $\\sum_{n=0}^\\infty b_n$", "converges absolutely to $AB.$" ], "refs": [], "proofs": [ { "contents": [ "Let $C_n$ be the $n$th partial sum of the Cauchy", "product; that is,", "$$", "C_n=c_0+c_1+\\cdots+c_n", "$$", "(see \\eqref{eq:4.3.38}). Let $\\sum_{n=0}^\\infty p_n$ be the series", "obtained", "by ordering the products $\\{a_i,b_j\\}$ according to the scheme", "indicated in \\eqref{eq:4.3.35}, and define $P_n$ to be its $n$th partial", "sum; thus,", "$$", "P_n=p_0+p_1+\\cdots+p_n.", "$$", "Inspection of \\eqref{eq:4.3.35} shows that $c_n$ is the sum of the $n+1$", "terms connected by the diagonal arrows. Therefore, $C_n=P_{m_n}$,", "where", "$$", "m_n=1+2+\\cdots+(n+1)-1=\\frac{n(n+3)}{2}.", "$$", "From Theorem~\\ref{thmtype:4.3.27}, $\\lim_{n\\to\\infty} P_{m_n}=AB$, so", "$\\lim_{n\\to\\infty} C_n=AB$. To see that $\\sum |c_n|<\\infty$, we", "observe that", "$$", "\\sum_{r=0}^n |c_r|\\le\\sum_{s=0}^{m_n} |p_s|", "$$", "\\nopagebreak", "and recall that $\\sum |p_s|<\\infty$, from Theorem~\\ref{thmtype:4.3.27}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.3.27", "TRENCH_REAL_ANALYSIS-thmtype:4.3.27" ], "ref_ids": [ 113, 113 ] } ], "ref_ids": [] }, { "id": 115, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.4", "categories": [], "title": "", "contents": [ "Let $\\{F_n\\}$ be defined on $S.$", "Then", "\\begin{alist}", "\\item % (a)", "$\\{F_n\\}$ converges pointwise to $F$ on $S$ if and only if there is,", "for each $\\epsilon>0$ and $x\\in S$, an integer $N$ $($which may depend", "on $x$ as well as $\\epsilon)$ such that", "$$", "|F_n(x)-F(x)|<\\epsilon\\mbox{\\quad if\\quad}\\ n\\ge N.", "$$", "\\item % (b)", " $\\{F_n\\}$ converges uniformly to $F$ on $S$ if and only if", "there is for each $\\epsilon>0$ an integer $N$ $($which depends only on", "$\\epsilon$ and not on any particular $x$ in $S)$ such that", "$$", "|F_n(x)-F(x)|<\\epsilon\\mbox{\\quad for all $x$ in $S$ if $n\\ge N$}.", "$$", "\\end{alist}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 116, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.5", "categories": [], "title": "", "contents": [ "If $\\{F_n\\}$ converges uniformly to $F$ on $S,$ then $\\{F_n\\}$ converges", "pointwise to $F$ on $S.$ The converse is false$;$ that is$,$ pointwise", "convergence does not imply uniform convergence." ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 117, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.6", "categories": [], "title": "Cauchy's Uniform Convergence Criterion", "contents": [ "A sequence of functions $\\{F_n\\}$ converges uniformly on a set $S$ if", "and", "only if for each $\\epsilon>0$ there is an integer $N$ such that", "\\begin{equation} \\label{eq:4.4.2}", "\\|F_n-F_m\\|_S<\\epsilon\\mbox{\\quad if\\quad} n, m\\ge N.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "For necessity, suppose that $\\{F_n\\}$ converges uniformly to", "$F$ on $S$. Then, if $\\epsilon>0$, there is an integer $N$ such that", "$$", "\\|F_k-F\\|_S<\\frac{\\epsilon}{2}\\mbox{\\quad if\\quad} k\\ge N.", "$$", "Therefore,", "\\begin{eqnarray*}", "\\|F_n-F_m\\|_S\\ar=\\|(F_n-F)+(F-F_m)\\|_S\\\\", "\\ar\\le \\|F_n-F\\|_S+\\|F-F_m\\|_S \\mbox{\\quad", "(Lemma~\\ref{thmtype:4.4.2})\\quad}\\\\", "&<&\\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}=\\epsilon\\mbox{\\quad if\\quad}", "m, n\\ge N.", "\\end{eqnarray*}", "For sufficiency, we first observe that \\eqref{eq:4.4.2} implies that", "$$", "|F_n(x)-F_m(x)|<\\epsilon\\mbox{\\quad if\\quad} n, m\\ge N,", "$$", "for any fixed $x$ in $S$. Therefore, Cauchy's convergence criterion", "for sequences of constants (Theorem~\\ref{thmtype:4.1.13})", "implies that", "$\\{F_n(x)\\}$ converges for each $x$ in $S$; that is, $\\{F_n\\}$", "converges pointwise to a limit function $F$ on $S$. To see that the", "convergence is uniform, we write", "\\begin{eqnarray*}", "|F_m(x)-F(x) |\\ar=|[F_m(x)-F_n(x)]+[F_n(x)-F(x)]|\\\\", "\\ar\\le |F_m(x)-F_n(x)|+| F_n(x)-F(x)|\\\\", "\\ar\\le \\|F_m-F_n\\|_S+|F_n(x)-F(x)|.", "\\end{eqnarray*}", "This and \\eqref{eq:4.4.2} imply that", "\\begin{equation} \\label{eq:4.4.3}", "|F_m(x)-F(x)|<\\epsilon+|F_n(x)-F(x)|\\quad\\mbox {if}\\quad n, m\\ge N.", "\\end{equation}", "Since $\\lim_{n\\to\\infty}F_n(x)=F(x)$,", "$$", "|F_n(x)-F(x)|<\\epsilon", "$$", "for some $n\\ge N$, so \\eqref{eq:4.4.3} implies that", "$$", "|F_m(x)-F(x)|<2\\epsilon\\mbox{\\quad if\\quad} m\\ge N.", "$$", "But this inequality holds for all $x$ in $S$, so", "$$", "\\|F_m-F\\|_S\\le2\\epsilon\\mbox{\\quad if\\quad} m\\ge N.", "$$", "Since $\\epsilon$ is an arbitrary positive number, this implies that", "$\\{F_n\\}$ converges uniformly to $F$ on~$S$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.4.2", "TRENCH_REAL_ANALYSIS-thmtype:4.1.13" ], "ref_ids": [ 251, 89 ] } ], "ref_ids": [] }, { "id": 118, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.7", "categories": [], "title": "", "contents": [ "If $\\{F_n\\}$ converges uniformly to $F$ on $S$ and each $F_n$ is", "continuous at a point $x_0$ in $S,$ then so is $F$. Similar", "statements hold for continuity from the right and left$.$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that each $F_n$ is continuous at $x_0$.", "If $x\\in S$ and $n\\ge1$, then", "\\begin{equation} \\label{eq:4.4.8}", "\\begin{array}{rcl}", "|F(x)-F(x_0)|\\ar\\le |F(x)-F_n(x)|+|F_n(x)-F_n(x_0)|+|F_n(x_0)-F(x_0)|", "\\\\", "\\ar\\le |F_n(x)-F_n(x_0)|+2\\|F_n-F\\|_S.", "\\end{array}", "\\end{equation}", "Suppose that $\\epsilon>0$. Since $\\{F_n\\}$ converges uniformly to $F$", "on $S$, we can choose $n$ so that $\\|F_n-F\\|_S<\\epsilon$. For this", "fixed $n$, \\eqref{eq:4.4.8} implies that", "\\begin{equation} \\label{eq:4.4.9}", "|F(x)-F(x_0)|<|F_n(x)-F_n(x_0)|+2\\epsilon,\\quad x\\in S.", "\\end{equation}", "Since $F_n$ is continuous at $x_0$, there is a $\\delta>0$ such that", "$$", "|F_n(x)-F_n(x_0)|<\\epsilon\\mbox{\\quad if\\quad} |x-x_0|<\\delta,", "$$", "so, from \\eqref{eq:4.4.9},", "$$", "|F(x)-F(x_0)|<3\\epsilon,\\mbox{\\quad if\\quad} |x-x_0|<\\delta.", "$$", "Therefore, $F$ is continuous at $x_0$. Similar", "arguments apply to the assertions on", "continuity from the right and left." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 119, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.9", "categories": [], "title": "", "contents": [ "Suppose that $\\{F_n\\}$ converges uniformly to $F$ on $S=[a,b]$. Assume", "that $F$ and all $F_n$", "are integrable on $[a,b].$ Then", "\\begin{equation} \\label{eq:4.4.10}", "\\int_a^b F(x)\\,dx=\\lim_{n\\to\\infty}\\int_a^b F_n(x)\\,dx.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Since", "\\begin{eqnarray*}", "\\left|\\int_a^b F_n(x)\\,dx-\\int_a^b F(x)\\,dx\\right|\\ar\\le \\int_a^b", "|F_n(x)-F(x)|\\,dx\\\\", "\\ar\\le (b-a)\\|F_n-F\\|_S", "\\end{eqnarray*}", "and $\\lim_{n\\to\\infty}\\|F_n-F\\|_S=0$, the conclusion follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 120, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.10", "categories": [], "title": "", "contents": [ " Suppose that $\\{F_n\\}$ converges", "pointwise to $F$ and each $F_n$ is integrable on $[a,b].$", "\\begin{alist}", "\\item % (a)", "If the convergence is uniform$,$ then $F$ is integrable on", "$[a,b]$ and $\\eqref{eq:4.4.10}$ holds.", "\\item % (b)", "If the sequence $\\{\\|F_n\\|_{[a,b]}\\}$ is bounded and $F$ is", "integrable on $[a,b],$ then $\\eqref{eq:4.4.10}$ holds.", "\\end{alist}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 121, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.11", "categories": [], "title": "", "contents": [ "Suppose that $F'_n$ is continuous on $[a,b]$ for all $n\\ge1$ and $\\{F'_n\\}$", "converges uniformly on $[a,b].$ Suppose also that", " $\\{F_n(x_0)\\}$ converges for some $x_0$ in $[a,b].$ Then", "$\\{F_n\\}$ converges uniformly on $[a,b]$ to a differentiable limit", "function $F,$ and", "\\begin{equation} \\label{eq:4.4.11}", "F'(x)=\\lim_{n\\to\\infty}F'_n(x),\\quad a0$ there is an integer $N$ such that", "\\vskip0pt", "\\begin{equation} \\label{eq:4.4.16}", "\\|f_n+f_{n+1}+\\cdots+f_m\\|_S<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge", "N.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Apply Theorem~\\ref{thmtype:4.4.6} to the partial sums of", "$\\sum f_n$, observing that", "$$", "f_n+f_{n+1}+\\cdots+f_m=F_m-F_{n-1}.", "$$", "\\vskip-2em" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.4.6" ], "ref_ids": [ 117 ] } ], "ref_ids": [] }, { "id": 123, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.15", "categories": [], "title": "Weierstrass's Test", "contents": [ "The series $\\sum f_n$ converges uniformly on $S$ if", "\\begin{equation} \\label{eq:4.4.17}", "\\|f_n\\|_S\\le M_n,\\quad n\\ge k,", "\\end{equation}", "where $\\sum M_n<\\infty.$" ], "refs": [], "proofs": [ { "contents": [ "From Cauchy's convergence criterion for series of constants,", "there is for each $\\epsilon>0$ an integer $N$ such that", "$$", "M_n+M_{n+1}+\\cdots+M_m<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N,", "$$", "which, because of \\eqref{eq:4.4.17}, implies that", "$$", "\\|f_n\\|_S+\\|f_{n+1}\\|_S+\\cdots+\\|f_m\\|_S<\\epsilon\\mbox{\\quad if\\quad}", " m, n\\ge N.", "$$", " Lemma~\\ref{thmtype:4.4.2} and Theorem~\\ref{thmtype:4.4.13} imply that", "$\\sum f_n$ converges uniformly on $S$.", "\\mbox{}" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.4.2", "TRENCH_REAL_ANALYSIS-thmtype:4.4.13" ], "ref_ids": [ 251, 122 ] } ], "ref_ids": [] }, { "id": 124, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.16", "categories": [], "title": "Dirichlet's Test for Uniform Convergence", "contents": [ "The series", "$$", "\\sum_{n=k}^\\infty f_ng_n", "$$", " converges uniformly on", "$S$ if", " $\\{f_n\\}$ converges uniformly to zero on $S,$", " $\\sum (f_{n+1}-f_n)$ converges absolutely uniformly on", "$S,$ and", "\\begin{equation} \\label{eq:4.4.19}", "\\|g_k+g_{k+1}+\\cdots+g_n\\|_S\\le M,\\quad n\\ge k,", "\\end{equation}", "for some constant $M.$" ], "refs": [], "proofs": [ { "contents": [ "The proof is similar to the proof of", "Theorem~\\ref{thmtype:4.3.20}. Let", "$$", "G_n=g_k+g_{k+1}+\\cdots+g_n,", "$$", "and consider the partial sums of $\\sum_{n=k}^\\infty f_ng_n$:", "\\begin{equation} \\label{eq:4.4.20}", "H_n=f_kg_k+f_{k+1}g_{k+1}+\\cdots+f_ng_n.", "\\end{equation}", "By substituting", "$$", "g_k=G_k\\mbox{\\quad and\\quad} g_n=G_n-G_{n-1},\\quad n\\ge k+1,", "$$", "into \\eqref{eq:4.4.20}, we obtain", "$$", "H_n=f_kG_k+f_{k+1}(G_{k+1}-G_k)+\\cdots+f_n(G_n-G_{n-1}),", "$$", "which we rewrite as", "$$", "H_n=(f_k-f_{k+1})", "G_k+(f_{k+1}-f_{k+2})G_{k+1}+\\cdots+(f_{n-1}-f_n)G_{n-1}+f_nG_n,", "$$", "or", "\\begin{equation} \\label{eq:4.4.21}", "H_n=J_{n-1}+f_nG_n,", "\\end{equation}", "where", "\\begin{equation} \\label{eq:4.4.22}", "J_{n-1}=(f_k-f_{k+1})G_k+(f_{k+1}-f_{k+2})", "G_{k+1}+\\cdots+(f_{n-1}-f_n)G_{n-1}.", "\\end{equation}", "That is, $\\{J_n\\}$ is the sequence of partial sums of the series", "\\begin{equation} \\label{eq:4.4.23}", "\\sum_{j=k}^\\infty (f_j-f_{j+1})G_j.", "\\end{equation}", " From \\eqref{eq:4.4.19} and the definition of", "$G_j$,", "$$", "\\left|\\sum^m_{j=n}[f_j(x)-f_{j+1}(x)]G_j(x)\\right|\\le M", "\\sum^m_{j=n}|f_j(x)-f_{j+1}(x)|,\\quad x\\in S,", "$$", "\\newpage", "\\noindent so", "$$", "\\left\\|\\sum^m_{j=n} (f_j-f_{j+1})G_j\\right\\|_S\\le M\\left\\|\\sum^m_{j=n}", "|f_j-f_{j+1}|\\right\\|_S.", "$$", "Now suppose that $\\epsilon>0$.", "Since $\\sum (f_j-f_{j+1})$ converges absolutely uniformly on $S$,", "Theorem~\\ref{thmtype:4.4.13} implies that", "there is an integer $N$ such that", "the right side of the last", "inequality is less than $\\epsilon$ if", "$m\\ge n\\ge N$. The same is then true of the left side, so", "Theorem~\\ref{thmtype:4.4.13}", " implies that", "\\eqref{eq:4.4.23} converges uniformly on~$S$.", "We have now shown that $\\{J_n\\}$ as defined in \\eqref{eq:4.4.22} converges", "uniformly to a limit function $J$ on $S$. Returning to \\eqref{eq:4.4.21},", "we see that", "$$", "H_n-J=J_{n-1}-J+f_nG_n.", "$$", "Hence, from Lemma~\\ref{thmtype:4.4.2} and \\eqref{eq:4.4.19},", "\\begin{eqnarray*}", "\\|H_n-J\\|_S\\ar\\le \\|J_{n-1}-J\\|_S+\\|f_n\\|_S\\|G_n\\|_S\\\\", "\\ar\\le \\|J_{n-1}-J\\|_S+M\\|f_n\\|_S.", "\\end{eqnarray*}", "Since $\\{J_{n-1}-J\\}$ and $\\{f_n\\}$ converge uniformly to zero on $S$,", "it now follows that $\\lim_{n\\to\\infty}\\|H_n-J\\|_S=0$. Therefore,", " $\\{H_n\\}$ converges uniformly on~$S$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.3.20", "TRENCH_REAL_ANALYSIS-thmtype:4.4.13", "TRENCH_REAL_ANALYSIS-thmtype:4.4.13", "TRENCH_REAL_ANALYSIS-thmtype:4.4.2" ], "ref_ids": [ 108, 122, 122, 251 ] } ], "ref_ids": [] }, { "id": 125, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.18", "categories": [], "title": "", "contents": [ "If $\\sum_{n=k}^\\infty f_n$ converges uniformly to $F$ on $S$ and each", "$f_n$ is continuous at a point $x_0$ in $S,$ then so is $F.$ Similar", "statements hold for continuity from the right and left$.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 126, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.19", "categories": [], "title": "", "contents": [ "Suppose that $\\sum_{n=k}^\\infty f_n$ converges uniformly to $F$ on", "$S=[a,b].$ Assume that $F$ and $f_n,$ $n\\ge k,$", "are integrable on $[a,b].$ Then", "$$", "\\int_a^b F(x)\\,dx=\\sum_{n=k}^\\infty \\int_a^b f_n(x)\\,dx.", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 127, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.20", "categories": [], "title": "", "contents": [ "Suppose that $f_n$ is continuously differentiable on $[a,b]$ for each", "$n\\ge k,$ $\\sum_{n=k}^\\infty f_n(x_0)$ converges for some $x_0$ in", "$[a,b],$ and", "$\\sum_{n=k}^\\infty f'_n$ converges uniformly on $[a,b].$ Then", "$\\sum_{n=k}^\\infty f_n$ converges uniformly on $[a,b]$ to a", "differentiable function $F,$ and", "$$", "F'(x)=\\sum_{n=k}^\\infty f'_n(x),\\quad aR.$ No general statement can be made concerning convergence", "at the endpoints $x=x_0+R$ and $x=x_0-R:$ the series may converge", "absolutely or conditionally at both$,$ converge conditionally at one", "and diverge at the other$,$ or diverge at both$.$" ], "refs": [], "proofs": [ { "contents": [ "In any case, the series \\eqref{eq:4.5.1} converges to $a_0$ if", "$x=x_0$. If", "\\begin{equation}\\label{eq:4.5.3}", "\\sum |a_n|r^n<\\infty", "\\end{equation}", "for some $r>0$, then $\\sum a_n (x-x_0)^n$ converges", "absolutely uniformly in $[x_0-r, x_0+r]$, by Weierstrass's test", "(Theorem~\\ref{thmtype:4.4.15}) and", "Exercise~\\ref{exer:4.4.21}. From Cauchy's root test", "(Theorem~\\ref{thmtype:4.3.17}),", "\\eqref{eq:4.5.3} holds if", "$$", "\\limsup_{n\\to\\infty} (|a_n|r^n)^{1/n}<1,", "$$", "which is equivalent to", " $$", " r\\,\\limsup_{n\\to\\infty} |a_n|^{1/n}<1", "$$", "(Exercise~\\ref{exer:4.1.30}\\part{a}).", "From \\eqref{eq:4.5.2}, this can be rewritten as $rR$, then", "\\newpage", "$$", "\\frac{1}{ R}>\\frac{1}{ |x-x_0|},", "$$", "so \\eqref{eq:4.5.2} implies that", "$$", "|a_n|^{1/n}\\ge\\frac{1}{ |x-x_0|}\\mbox{\\quad and therefore\\quad}", "|a_n(x-x_0)^n|\\ge1", "$$", "for infinitely many values of $n$. Therefore, $\\sum a_n(x-x_0)^n$", "diverges (Corollary~\\ref{thmtype:4.3.6}) if $|x-x_0|>R$.", "In particular, the series diverges for all $x\\ne x_0$ if $R=0$.", "To prove the assertions concerning the possibilities at $x=x_0+R$", "and $x=x_0-R$ requires examples, which follow. (Also, see", "Exercise~\\ref{exer:4.5.1}.)" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.4.15", "TRENCH_REAL_ANALYSIS-thmtype:4.3.17", "TRENCH_REAL_ANALYSIS-thmtype:4.3.6" ], "ref_ids": [ 123, 106, 277 ] } ], "ref_ids": [] }, { "id": 129, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.5.3", "categories": [], "title": "", "contents": [ "The radius of convergence of $\\sum", "a_n(x-x_0)^n$ is given by", "$$", "\\frac{1}{ R}=\\lim_{n\\to\\infty}\\left|\\frac{a_{n+1}}{a_n}\\right|", "$$", "if the limit exists in the extended reals$.$" ], "refs": [], "proofs": [ { "contents": [ "From Theorem~\\ref{thmtype:4.5.2}, it suffices to show that if", "\\begin{equation}\\label{eq:4.5.4}", "L=\\lim_{n\\to\\infty}\\left|\\frac{a_{n+1}}{a_n}\\right|", "\\end{equation}", "exists in the extended reals, then", "\\begin{equation}\\label{eq:4.5.5}", "L=\\limsup_{n\\to\\infty}|a_n|^{1/n}.", "\\end{equation}", "We will show that this is so if $0 N.", "$$", "Therefore, if", "$$", "K_1=|a_N|(L-\\epsilon)^{-N}\\mbox{\\quad and\\quad} K_2=|a_N|(L+", "\\epsilon)^{-N},", "$$", "then", "\\begin{equation}\\label{eq:4.5.6}", "K^{1/n}_1(L-\\epsilon)<|a_n|^{1/n}0$, choose $N$ so that", "$$", "|s_n-s|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N+1.", "$$", "Then, if $00$. If", "$$", "|\\mathbf{X}_1-\\mathbf{X}_0|0$ there is an integer $K$ such that", "$$", "|\\mathbf{X}_r-\\mathbf{X}_s|<\\epsilon\\mbox{\\quad if\\quad} r,s\\ge K.", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 142, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.17", "categories": [], "title": "Principle of Nested Sets", "contents": [ "If $S_1,$ $S_2,$ \\dots\\ are closed nonempty subsets of $\\R^n$", "such that", "\\begin{equation}\\label{eq:5.1.14}", "S_1\\supset S_2\\supset\\cdots\\supset S_r\\supset\\cdots", "\\end{equation}", "and", "\\begin{equation}\\label{eq:5.1.15}", "\\lim_{r\\to\\infty} d(S_r)=0,", "\\end{equation}", "then the intersection", "$$", "I=\\bigcap^\\infty_{r=1}S_r", "$$", "contains exactly one point$.$" ], "refs": [], "proofs": [ { "contents": [ "Let", "$\\{\\mathbf{X}_r\\}$ be a sequence such that $\\mathbf{X}_r\\in S_r\\ (r\\ge1)$.", "Because of", "\\eqref{eq:5.1.14}, $\\mathbf{X}_r\\in S_k$ if $r\\ge k$, so", "$$", "|\\mathbf{X}_r-\\mathbf{X}_s|2$. The counterpart of the", "square $T$ is the {\\it hypercube\\/} with sides of", "length", "$L$:", "$$", "T=\\set{(x_1,x_2, \\dots,x_n)}{ a_i\\le x_i\\le a_i+L, i=1,2, \\dots, n}.", "$$", "Halving the intervals of variation of the $n$ coordinates", "$x_1$, $x_2$, \\dots, $x_n$ divides $T$ into $2^n$ closed hypercubes", "with sides of length $L/2$:", "$$", "T^{(i)}=\\set{(x_1,x_2, \\dots,x_n)}{b_i\\le x_i\\le b_i+L/2,", "1\\le i\\le n},", "$$", "where $b_i=a_i$ or $b_i=a_i+L/2$. If no finite subcollection of ${\\mathcal", "H}$ covers $S$, then at least one of these smaller hypercubes must", "contain a subset of $S$ that is not covered by any finite subcollection", "of $S$. Now the proof proceeds as for $n=2$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.1.17" ], "ref_ids": [ 142 ] } ], "ref_ids": [] }, { "id": 144, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.20", "categories": [], "title": "", "contents": [ " An open set $S$ in $\\R^n$ is", "connected if and only if it is polygonally connected$.$" ], "refs": [], "proofs": [ { "contents": [ "For sufficiency, we will show that if $S$ is disconnected, then", "$S$ is not poly\\-gonally connected. Let $S=A\\cup B$, where $A$ and $B$", "satisfy \\eqref{eq:5.1.16}. Suppose that $\\mathbf{X}_1\\in A$ and $\\mathbf{X}_2\\in", "B$, and assume that there is a polygonal path in $S$ connecting", "$\\mathbf{X}_{1}$ to $\\mathbf{X}_2$. Then some line segment $L$ in this", "path must", "contain a point $\\mathbf{Y}_1$ in $A$ and a point $\\mathbf{Y}_2$ in $B$. The", "line segment", "$$", "\\mathbf{X}=t\\mathbf{Y}_2+(1-t)\\mathbf{Y}_1,\\quad 0\\le t\\le1,", "$$", "is part of $L$ and therefore in $S$. Now define", "$$", "\\rho=\\sup\\set{\\tau}{ tY_2+(1-t)\\mathbf{Y}_1\\in A,\\ 0\\le t\\le", "\\tau\\le1},", "$$", "and let", "$$", "\\mathbf{X}_\\rho=\\rho\\mathbf{Y}_2+(1-\\rho)\\mathbf{Y}_1.", "$$", "Then $\\mathbf{X}_\\rho\\in\\overline{A}\\cap\\overline{B}$. However, since", "$\\mathbf{X}_\\rho\\in A\\cup B $ and $\\overline{A}\\cap", "B=A\\cap\\overline{B}=\\emptyset$, this is impossible. Therefore,", "the assumption that there is a polygonal path in $S$", "from $\\mathbf{X}_1$ to $\\mathbf{X}_2$ must be false.", "For necessity, suppose that $S$ is a connected open set and $\\mathbf{X}_0\\in", "S$. Let $A$ be the set consisting of $\\mathbf{X}_0$ and the points in $S$", "can be connected to $\\mathbf{X}_0$ by polygonal paths in $S$. Let $B$ be", "set of points in $S$ that cannot be connected to $\\mathbf{X}_0$", "by polygonal paths.", " If $\\mathbf{Y}_0\\in S$, then $S$ contains an", "$\\epsilon$-neighborhood $N_\\epsilon (\\mathbf{Y}_0)$ of $\\mathbf{Y}_0$,", "since $S$ is open. Any point $\\mathbf{Y}_1$ in $N_\\epsilon", "(\\mathbf{Y}_{0}$", " can be connected to $\\mathbf{Y}_0$ by the line segment", "$$", "\\mathbf{X}=t\\mathbf{Y}_1+(1-t)\\mathbf{Y}_0,\\quad 0\\le t\\le1,", "$$", "which lies in $N_\\epsilon(\\mathbf{Y}_0)$ (Lemma~\\ref{thmtype:5.1.12}) and", "therefore in", "$S$. This implies that $\\mathbf{Y}_0$ can be connected to $\\mathbf{X}_0$ by a", "polygonal path in $S$ if and only if every member of $N_\\epsilon", "(\\mathbf{Y}_{0})$", " can also. Thus, $N_\\epsilon(\\mathbf{Y}_0)\\subset A$ if $\\mathbf{Y}_0\\in", "A$, and $N_\\epsilon (\\mathbf{Y}_0)\\in B$ if $\\mathbf{Y}_0\\in B$. Therefore,", "$A$ and $B$ are open. Since $A\\cap B =\\emptyset$, this implies that", "$A\\cap\\overline{B}=\\overline{A}\\cap B=\\emptyset$", "(Exercise~\\ref{exer:5.1.14}). Since $A$ is nonempty $(\\mathbf{X}_0\\in A)$,", "it", "now follows that $B=\\emptyset$, since if $B\\ne\\emptyset$, $S$ would be", "disconnected (Definition~\\ref{thmtype:5.1.19}). Therefore, $A=S$, which", "completes the proof of necessity." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.1.12", "TRENCH_REAL_ANALYSIS-thmtype:5.1.19" ], "ref_ids": [ 253, 343 ] } ], "ref_ids": [] }, { "id": 145, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.2", "categories": [], "title": "", "contents": [ " If $\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})$ exists$,$ then it is", "unique." ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 146, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.3", "categories": [], "title": "", "contents": [ "Suppose that $f$ and $g$ are defined on a set $D,$ $\\mathbf{X}_0$ is a", "limit point of $D,$ and", "$$", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=L_1,\\quad\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} g(\\mathbf{X})=L_2.", "$$", "Then", "\\begin{eqnarray}", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}(f+g)(\\mathbf{X})\\ar=L_1+L_2,\\label{eq:5.2.10}\\\\", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}(f-g)(\\mathbf{X})\\ar=L_1-L_2,\\label{eq:5.2.11}\\\\", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}(fg)(\\mathbf{X})\\ar=L_1L_2,\\label{eq:5.2.12}\\\\", "\\arraytext{and$,$ if $L_2\\ne0,$}\\nonumber\\\\", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}\\left(\\frac{f}{ g}\\right)(\\mathbf{X})", "\\ar=\\frac{L_1}{ L_2}.\\label{eq:5.2.13}", "\\end{eqnarray}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 147, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.7", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{X}_0$ is in $D_f$ and is a limit point of $D_f.$ Then", "$f$", "is continuous at $\\mathbf{X}_0$ if and only if for each $\\epsilon>0$ there", "is a $\\delta>0$ such that", "$$", "\\left|f(\\mathbf{X})-f(\\mathbf{X}_0)\\right|<\\epsilon", "$$", "whenever", "$$", "|\\mathbf{X}-\\mathbf{X}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{X}\\in D_f.", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 148, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.8", "categories": [], "title": "", "contents": [ "If $f$ and $g$ are continuous on a set $S$ in $\\R^n,$ then so", "are $f+g,$ $f-g,$ and $fg.$ Also$,$ $f/g$ is continuous at each", "$\\mathbf{X}_0$ in $S$ such that $g(\\mathbf{X}_0)\\ne0.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 149, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.9", "categories": [], "title": "", "contents": [ "For a vector-valued function $\\mathbf{G},$", "$$", "\\lim_{\\mathbf{U}\\to\\mathbf{U}_0}\\mathbf{G}(\\mathbf{U})=\\mathbf{L}", "$$", "if and only if for each $\\epsilon>0$ there is a $\\delta>0$ such that", "$$", "|\\mathbf{G}(\\mathbf{U})-\\mathbf{L}|<\\epsilon\\mbox{\\quad whenever\\quad}", "0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{U}\\in D_{\\mathbf{G}}.", "$$", "Similarly, $\\mathbf{G}$ is continuous at $\\mathbf{U}_0$ if and only if for", "each", "$\\epsilon> 0$ there is a $\\delta>0$ such that", "$$", "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|<\\epsilon", "\\mbox{\\quad whenever\\quad}", " |\\mathbf{U}-\\mathbf{U}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{U}\\in D_{\\mathbf{G}}.", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 150, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.10", "categories": [], "title": "", "contents": [ "Let $f$ be a real-valued function defined on a subset of $\\R^n,$", " and let the", "vector-valued function $\\mathbf{G}=(g_1,g_2, \\dots,g_n)$ be defined on a", "domain $D_\\mathbf{G}$ in $\\R^m.$ Let the set", "$$", "T=\\set{\\mathbf{U}}{\\mathbf{U}\\in D_{\\mathbf{G}}\\mbox{\\quad and \\quad}", "\\mathbf{G}(\\mathbf{U})\\in D_f}", "$$", "$($Figure~\\ref{figure:5.2.3}$)$,", " be", "nonempty$,$ and define the real-valued composite function", "$$", "h=f\\circ\\mathbf{G}", "$$", "on $T$ by", "$$", "h(\\mathbf{U})=f(\\mathbf{G}(\\mathbf{U})),\\quad \\mathbf{U}\\in T.", "$$", "Now suppose that $\\mathbf{U}_0$ is in $T$ and is a limit point of $T,$", "$\\mathbf{G}$ is continuous at $\\mathbf{U}_0,$ and $f$ is continuous at", "$\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0).$ Then $h$ is continuous at", "$\\mathbf{U}_0.$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\epsilon>0$. Since $f$ is continuous at", "$\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0)$, there is an $\\epsilon_1>0$", "such that", "\\begin{equation}\\label{eq:5.2.17}", "|f(\\mathbf{X})-f(\\mathbf{G}(\\mathbf{U}_0))|<\\epsilon", "\\end{equation}", "if", "\\begin{equation}\\label{eq:5.2.18}", "|\\mathbf{X}-\\mathbf{G}(\\mathbf{U}_0)|<\\epsilon_1\\mbox{\\quad and\\quad}", "\\mathbf{X}\\in D_f.", "\\end{equation}", "Since $\\mathbf{G}$ is continuous at $\\mathbf{U}_0$, there is a $\\delta>0$", "such that", "$$", "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|<\\epsilon_1", "\\mbox{\\quad if\\quad} |\\mathbf{U}-\\mathbf{U}_0|<", "\\delta\\mbox{\\quad and\\quad}\\mathbf{U}\\in D_\\mathbf{G}.", "$$", "By taking $\\mathbf{X}=\\mathbf{G}(\\mathbf{U})$ in \\eqref{eq:5.2.17} and", "\\eqref{eq:5.2.18}, we see that", "$$", "|h(\\mathbf{U})-h(\\mathbf{U}_0)|=|f(\\mathbf{G}(\\mathbf{U})", "-f(\\mathbf{G}(\\mathbf{U}_0))|<\\epsilon", "$$", "if", "$$", "|\\mathbf{U}-\\mathbf{U}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{U}\\in T.", "$$" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 151, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.11", "categories": [], "title": "", "contents": [ "If $f$ is continuous on a compact set $S$ in $\\R^n,$ then $f$", "is bounded on~$S.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 152, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.12", "categories": [], "title": "", "contents": [ "Let $f$ be continuous on a compact set $S$ in $\\R^n$ and", "$$", "\\alpha=\\inf_{\\mathbf{X}\\in S}f(\\mathbf{X}),\\quad\\beta=", "\\sup_{\\mathbf{X}\\in S}f(\\mathbf{X}).", "$$", "Then", "$$", "f(\\mathbf{X}_1)=\\alpha\\mbox{\\quad and\\quad} f(\\mathbf{X}_2)=\\beta", "$$", "for some $\\mathbf{X}_1$ and $\\mathbf{X}_2$ in $S.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 153, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.13", "categories": [], "title": "Intermediate Value Theorem", "contents": [ "Let $f$ be continuous on a region $S$ in $\\R^n.$ Suppose that", "$\\mathbf{A}$ and $\\mathbf{B}$ are in $S$ and", "$$", "f(\\mathbf{A})u}.", "\\end{eqnarray*}", "If $\\mathbf{X}_0\\in R$, the continuity of $f$ implies that there is a", "$\\delta>0$ such that $f(\\mathbf{X})0$. Choose $\\delta>0$ so that", "the open square", "\\newpage", "$$", "S_\\delta=\\set{(x,y)}{|x-x_0|<\\delta, |y-y_0|<\\delta}", "$$", "is in $N$ and", "\\begin{equation}\\label{eq:5.3.6}", "|f_{xy}(\\widehat{x},\\widehat{y})-f_{xy}(x_0,y_0)|<\\epsilon\\quad", "\\mbox{\\quad if\\quad}(\\widehat{x},\\widehat{y})\\in S_\\delta.", "\\end{equation}", "This is possible because of the continuity of $f_{xy}$ at $(x_0,y_0)$.", "The function", "\\begin{equation}\\label{eq:5.3.7}", "A(h,k)=f(x_0+h, y_0+k)-f(x_0+h,y_0)-f(x_0,y_0+k)+f(x_0,y_0)", "\\end{equation}", "is defined if $-\\delta0$. Our assumptions imply that there is", "a $\\delta>0$ such that $f_{x_1}, f_{x_2}, \\dots, f_{x_n}$ are defined", "in the $n$-ball", "$$", "S_\\delta (\\mathbf{X}_0)=\\set{\\mathbf{X}}{|\\mathbf{X}-\\mathbf{X}_0|<\\delta}", "$$", "and", "\\begin{equation}\\label{eq:5.3.24}", "|f_{x_j}(\\mathbf{X})-f_{x_j}(\\mathbf{X}_0)|<\\epsilon\\mbox{\\quad if\\quad}", "|\\mathbf{X}-\\mathbf{X}_0|<\\delta,\\quad 1\\le j\\le n.", "\\end{equation}", "Let $\\mathbf{X}=(x_1,x_, \\dots,x_n)$ be in $S_\\delta(\\mathbf{X}_0)$.", "Define", "$$", "\\mathbf{X}_j=(x_1, \\dots,x_j, x_{j+1,0}, \\dots,x_{n0}),\\quad 1\\le j\\le n-1,", "$$", "and", "$\\mathbf{X}_n=\\mathbf{X}$.", "Thus, for $1\\le j\\le n$, $\\mathbf{X}_j$ differs from $\\mathbf{X}_{j-1}$", " in the", "$j$th component only, and the line segment from $\\mathbf{X}_{j-1}$ to", "$\\mathbf{X}_j$ is in $S_\\delta (\\mathbf{X}_0)$.", "Now write", "\\begin{equation}\\label{eq:5.3.25}", "f(\\mathbf{X})-f(\\mathbf{X}_0)=f(\\mathbf{X}_n)-f(\\mathbf{X}_0)=", "\\sum^n_{j=1}\\,[f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})],", "\\end{equation}", "and consider the auxiliary functions", "\\begin{equation}\\label{eq:5.3.26}", "\\begin{array}{rcl}", "g_1(t)\\ar=f(t,x_{20}, \\dots,x_{n0}),\\\\[2\\jot]", "g_j(t)\\ar=f(x_1, \\dots,x_{j-1},t,x_{j+1,0}, \\dots,x_{n0}),\\quad 2\\le j\\le", "n-1,\\\\[2\\jot]", "g_n(t)\\ar=f(x_1, \\dots,x_{n-1},t),", "\\end{array}", "\\end{equation}", "where, in each case, all variables except $t$ are temporarily regarded", "as constants. Since", "$$", "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=g_j(x_j)-g_j(x_{j0}),", "$$", "the mean value theorem implies that", "$$", "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=g'_j(\\tau_j)(x_j-x_{j0}),", "$$", "\\newpage", "\\noindent", "where $\\tau_j$ is between $x_j$ and $x_{j0}$. From \\eqref{eq:5.3.26},", "$$", "g'_j(\\tau_j)=f_{x_j}(\\widehat{\\mathbf{X}}_j),", "$$", "where $\\widehat{\\mathbf{X}}_j$ is on the line segment from $\\mathbf{X}_{j-1}$ to", "$\\mathbf{X}_j$. Therefore,", "$$", "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=f_{x_j}(\\widehat{\\mathbf{X}}_j)(x_j-x_{j0}),", "$$", "and \\eqref{eq:5.3.25} implies that", "\\begin{eqnarray*}", "f(\\mathbf{X})-f(\\mathbf{X}_0)\\ar=\\sum^n_{j=1} f_{x_j} (\\widehat{\\mathbf{X}}_j)(x_j-x_{j0})\\\\", "\\ar=\\sum^n_{j=1} f_{x_j}(\\mathbf{X}_0) (x_j-x_{j0})+\\sum^n_{j=1}", "\\,[f_{x_j}(\\widehat{\\mathbf{X}}_j)-f_{x_j}(\\mathbf{X}_0)](x_j-x_{j0}).", "\\end{eqnarray*}", "From this and \\eqref{eq:5.3.24},", "$$", "\\left|f(\\mathbf{X})-f(\\mathbf{X}_0)-\\sum^n_{j=1}", "f_{x_j}(\\mathbf{X}_{0})", "(x_j-x_{j0})\\right|\\le", "\\epsilon\\sum^n_{j=1} |x_j-x_{j0}|\\le n\\epsilon |\\mathbf{X}-\\mathbf{X}_0|,", "$$", "which implies that $f$ is differentiable at $\\mathbf{X}_0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 162, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.3.11", "categories": [], "title": "", "contents": [ "Suppose that $f$ is defined in a neighborhood of $\\mathbf{X}_0$ in", "$\\R^n$ and $f_{x_1}(\\mathbf{X}_0),$ $f_{x_2}(\\mathbf{X}_{0}),$", " \\dots$,$ $f_{x_n}(\\mathbf{X}_{0})$", " exist$.$ Let $\\mathbf{X}_0$ be a local extreme point of $f.$ Then", "\\begin{equation}\\label{eq:5.3.42}", "f_{x_i}(\\mathbf{X}_0)=0,\\quad 1\\le i\\le n.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Let", "$$", "\\mathbf{E}_1=(1,0, \\dots,0),\\quad \\mathbf{E}_{2}", "=(0,1,0, \\dots,0),\\dots,\\quad \\mathbf{E}_n=", "(0,0, \\dots,1),", "$$", "and", "$$", "g_i(t)=f(\\mathbf{X}_0+t\\mathbf{E}_i),\\quad 1\\le i\\le n.", "$$", "Then $g_i$ is differentiable at $t=0$, with", "$$", "g'_i(0)=f_{x_i}(\\mathbf{X}_0)", "$$", "\\newpage", "\\noindent", "(Definition~\\ref{thmtype:5.3.1}). Since $\\mathbf{X}_0$ is a local extreme", "point of $f$, $t_0=0$ is a local extreme point of $g_i$. Now", "Theorem~\\ref{thmtype:2.3.7} implies that $g'_i(0)=0$, and this", "implies \\eqref{eq:5.3.42}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.3.1", "TRENCH_REAL_ANALYSIS-thmtype:2.3.7" ], "ref_ids": [ 349, 31 ] } ], "ref_ids": [] }, { "id": 163, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.3", "categories": [], "title": "The Chain Rule", "contents": [ "Suppose that the real-valued function $f$ is differentiable at", "$\\mathbf{X}_0$", "in $\\R^n,$ the vector-valued function $\\mathbf{G}", "=(g_1,g_2, \\dots,g_n)$ is differentiable at", "$\\mathbf{U}_0$ in $\\R^m,$ and $\\mathbf{X}_{0}", " = \\mathbf{G}(\\mathbf{U}_0).$ Then the real-valued composite function", "$h=f\\circ\\mathbf{G}$ defined by", "\\begin{equation} \\label{eq:5.4.3}", "h(\\mathbf{U})=f(\\mathbf{G}(\\mathbf{U}))", "\\end{equation}", "is differentiable at $\\mathbf{U}_0,$ and", "\\begin{equation} \\label{eq:5.4.4}", "d_{\\mathbf{U}_0}h=f_{x_1}(\\mathbf{X}_0) d_{\\mathbf{U}_0}g_1+f_{x_2}", "(\\mathbf{X}_0) d_{\\mathbf{U}_0}g_2+\\cdots", "+f_{x_n} (\\mathbf{X}_0) d_{\\mathbf{U}_0}g_n.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "We leave it to you to show that $\\mathbf{U}_0$ is an interior point", "of the domain of $h$ (Exercise~\\ref{exer:5.4.1}), so it is legitimate to", "ask if $h$ is differentiable at $\\mathbf{U}_0$.", "Let $\\mathbf{X}_0=(x_{10},x_{20}, \\dots,x_{n0})$. Note that", "$$", "x_{i0}=g_i(\\mathbf{U}_0),\\quad", "1\\le i\\le n,", "$$", "by assumption.", "Since $f$ is differentiable at $\\mathbf{X}_0$,", "Lemma~\\ref{thmtype:5.3.8} implies that", "\\begin{equation} \\label{eq:5.4.5}", "f(\\mathbf{X})-f(\\mathbf{X}_0)=\\sum_{i=1}^n f_{x_i} (\\mathbf{X}_0)", "(x_i-x_{i0})+E(\\mathbf{X})|\\mathbf{X}-\\mathbf{X}_0|,", "\\end{equation}", "where", "$$", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}E(\\mathbf{X})=0.", "$$", "\\newpage", "\\noindent", " Substituting $\\mathbf{X}=\\mathbf{G}(\\mathbf{U})$", " and $\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0)$ in \\eqref{eq:5.4.5} and recalling", "\\eqref{eq:5.4.3} yields", "\\begin{equation} \\label{eq:5.4.6}", "h(\\mathbf{U})-h(\\mathbf{U}_0)=\\dst{\\sum_{i=1}^n}\\, f_{x_i}(\\mathbf{X}_0)", "(g_i(\\mathbf{U})-g_i(\\mathbf{U}_0))", "+E(\\mathbf{G}(\\mathbf{U}))", "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|.", "\\end{equation}", "Substituting \\eqref{eq:5.4.1} into \\eqref{eq:5.4.6} yields", "$$", "\\begin{array}{rcl}", "h(\\mathbf{U})-h(\\mathbf{U}_0)\\ar=\\dst{\\sum_{i=1}^n} f_{x_i}(\\mathbf{X}_0)", "(d_{\\mathbf{U}_0}g_i) (\\mathbf{U}-\\mathbf{U}_0)", "+\\dst{\\left(\\sum_{i=1}^n", "f_{x_i}(\\mathbf{X}_0)E_i(\\mathbf{U})\\right)} |\\mathbf{U}-\\mathbf{U}_0|", "\\\\\\\\", "\\ar{}+E(\\mathbf{G}(\\mathbf{U}))", "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_{0}|.", "\\end{array}", "$$", "Since", "$$", "\\lim_{\\mathbf{U}\\to\\mathbf{U}_0}E(\\mathbf{G}(\\mathbf{U}))=\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}E(\\mathbf{X})=0,", "$$", "\\eqref{eq:5.4.2} and Lemma~\\ref{thmtype:5.4.2} imply that", "$$", "\\frac{h(\\mathbf{U})-h(\\mathbf{U}_0)-\\dst\\sum_{i=1}^nf_{x_i}(\\mathbf{X}_{0}", "d_{\\mathbf{U}_0}g_i", "(\\mathbf{U}-\\mathbf{U}_0)}{|\\mathbf{U}-\\mathbf{U}_0|}=0.", "$$", "Therefore, $h$ is differentiable at $\\mathbf{U}_0$, and $d_{\\mathbf{U}_0}h$", "is given by \\eqref{eq:5.4.4}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.3.8", "TRENCH_REAL_ANALYSIS-thmtype:5.4.2" ], "ref_ids": [ 254, 255 ] } ], "ref_ids": [] }, { "id": 164, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.5", "categories": [], "title": "Mean Value Theorem for Functions of $\\mathbf n$ Variables", "contents": [ "Let $f$ be continuous at $\\mathbf{X}_1=(x_{11},x_{21}, \\dots, x_{n1})$", "and $\\mathbf{X}_2=(x_{12},x_{22}, \\dots,x_{n2})$ and differentiable on the", "line segment $L$ from $\\mathbf{X}_1$ to $\\mathbf{X}_2.$ Then", "\\begin{equation} \\label{eq:5.4.21}", "f(\\mathbf{X}_2)-f(\\mathbf{X}_1)=\\sum_{i=1}^n f_{x_i} (\\mathbf{X}_0)(x_{i2}-x_{i1})=(d_{\\mathbf{X}_0}f)(\\mathbf{X}_2", "-\\mathbf{X}_1)", "\\end{equation}", "for some $\\mathbf{X}_0$ on $L$ distinct", "from $\\mathbf{X}_1$ and $\\mathbf{X}_2$." ], "refs": [], "proofs": [ { "contents": [ "An equation of $L$ is", "$$", "\\mathbf{X}=\\mathbf{X}(t)=t\\mathbf{X}_2+(1-t)\\mathbf{X}_1,\\quad 0\\le t\\le1.", "$$", "Our hypotheses imply that the function", "$$", "h(t)=f(\\mathbf{X}(t))", "$$", "is continuous on $[0,1]$ and differentiable on $(0,1)$. Since", "$$", "x_i(t)=tx_{i2}+(1-t)x_{i1},", "$$", "\\eqref{eq:5.4.20} implies that", "$$", "h'(t)=\\sum_{i=1}^n f_{x_i}(\\mathbf{X}(t))(x_{i2}-x_{i1}),\\quad 00$, there is a $\\delta>0$ such that", "$B_\\delta (\\mathbf{X}_0)\\subset N$ and all $k$th-order partial", "derivatives of $f$ satisfy the inequality", "\\begin{equation} \\label{eq:5.4.32}", "\\left|\\frac{\\partial^kf(\\widetilde{\\mathbf{X}})}{\\partial x_{i_k}\\partial", "x_{i_{k-1}} \\cdots\\partial x_{i_1}}-", "\\frac{\\partial^kf(\\mathbf{X}_0)}{\\partial x_{i_k} \\partial", "x_{i_{k-1}}\\cdots\\partial", "x_{i_1}}\\right|<\\epsilon,\\quad \\widetilde{\\mathbf{X}}\\in B_\\delta (\\mathbf{X}_0).", "\\end{equation}", " Now suppose that $\\mathbf{X}\\in B_\\delta (\\mathbf{X}_0)$. From", "Theorem~\\ref{thmtype:5.4.8} with $k$ replaced by $k-1$,", "\\begin{equation} \\label{eq:5.4.33}", "f(\\mathbf{X})=T_{k-1}(\\mathbf{X})+\\frac{1}{ k!}", "(d^{(k)}_{\\widetilde{\\mathbf{X}}} f)(\\mathbf{X}-\\mathbf{X}_0),", "\\end{equation}", "where $\\widetilde{\\mathbf{X}}$ is some point", " on the line segment from $\\mathbf{X}_0$ to $\\mathbf{X}$ and is therefore", "in $B_\\delta(\\mathbf{X}_0)$. We can rewrite \\eqref{eq:5.4.33} as", "\\begin{equation} \\label{eq:5.4.34}", " f(\\mathbf{X})=T_k(\\mathbf{X})+\\frac{1}{", "k!}\\left[(d^{(k)}_{\\widetilde{\\mathbf{X}}} f)(\\mathbf{X}-\\mathbf{X}_0)-", "(d^{(k)}_{\\mathbf{X}_0} f)(\\mathbf{X}-\\mathbf{X}_0)\\right].", "\\end{equation}", "But \\eqref{eq:5.4.23} and", "\\eqref{eq:5.4.32} imply that", "\\begin{equation} \\label{eq:5.4.35}", "\\left|(d^{(k)}_{\\widetilde{\\mathbf{X}}}f)(\\mathbf{X}-\\mathbf{X}_0)-(d^{(k)}_{{\\mathbf{X}}_0}f)(\\mathbf{X}-\\mathbf{X}_0)\\right|< n^k\\epsilon |\\mathbf{X}-\\mathbf{X}_0|^k", "\\end{equation}", " (Exercise~\\ref{exer:5.4.17}), which", "implies that", "$$", "\\frac{|f(\\mathbf{X})-T_k(\\mathbf{X})|}", "{ |\\mathbf{X}-\\mathbf{X}_0|^k}<\\frac{n^k\\epsilon}{ k!}, \\quad\\mathbf{X}\\in", "B_\\delta (\\mathbf{X}_0),", "$$", "from \\eqref{eq:5.4.34}.", "This implies \\eqref{eq:5.4.31}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.4.8" ], "ref_ids": [ 165 ] } ], "ref_ids": [] }, { "id": 167, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.10", "categories": [], "title": "", "contents": [ "Suppose that $f$ satisfies the hypotheses of Theorem~$\\ref{thmtype:5.4.9}$", "with $k\\ge2,$ and", " \\begin{equation} \\label{eq:5.4.38}", "d^{(r)}_{\\mathbf{X}_0} f\\equiv0\\quad (1\\le r\\le k-1),\\quad d^{(k)}_\\mathbf{X_0}", "f\\not\\equiv0.", "\\end{equation}", "Then", "\\begin{alist}", "\\item % (a)", "$\\mathbf{X}_0$ is not a local extreme point of $f$ unless $d^{(k)}_{\\mathbf{X}_0}f$ is semidefinite as a polynomial in $\\mathbf{X}-\\mathbf{X}_0.$", "In particular$,$", " $\\mathbf{X}_0$ is not a local extreme point of $f$ if", "$k$ is odd$.$", "\\item % (b)", " $\\mathbf{X}_0$ is a local minimum point of $f$ if $d^{(k)}_{\\mathbf{X}_0}", "f$ is positive definite$,$ or a local maximum point if $d^{(k)}_{\\mathbf{X}_0}f$ is", "negative definite$.$", "\\item % (c)", " If $d^{(k)}_{\\mathbf{X}_0}f$ is semidefinite$,$ then $\\mathbf{X}_0$ may be a", "local extreme point of $f,$ but it need not be$.$", "\\end{alist}" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.4.9" ], "proofs": [ { "contents": [ "From \\eqref{eq:5.4.38} and Theorem~\\ref{thmtype:5.4.9},", "\\begin{equation} \\label{eq:5.4.39}", "\\lim_{ \\mathbf{X}\\to\\mathbf{X}_0}", "\\frac{f(\\mathbf{X})-f(\\mathbf{X}_0)-\\dst\\frac{1}{k!}", "(d^{(k)}_{\\mathbf{X}_0})(\\mathbf{X}-\\mathbf{X}_0)}{ |\\mathbf{X}-\\mathbf{X}_0|^k}=0.", "\\end{equation}", "If $\\mathbf{X}=\\mathbf{X}_0+t\\mathbf{U}$, where $\\mathbf{U}$ is a constant", "vector, then", "$$", "(d^{(k)}_{\\mathbf{X}_0} f) (\\mathbf{X}-\\mathbf{X}_0)=", "t^k(d^{(k)}_{\\mathbf{X}_0} f)(\\mathbf{U}),", "$$", "so \\eqref{eq:5.4.39} implies that", "$$", "\\lim_{t\\to 0} \\frac{f(\\mathbf{X}_0+t\\mathbf{U})-", "f(\\mathbf{X}_0)-\\dst\\frac{t^k}{k!}(d^{(k)}_{\\mathbf{X}_0}f)(\\mathbf{U})}{", "t^k}=0,", "$$", "or, equivalently,", "\\begin{equation} \\label{eq:5.4.40}", "\\lim_{t\\to 0}\\frac{f(\\mathbf{X}_0+t\\mathbf{U})-f(\\mathbf{X}_0)}{ t^k}=\\frac{1}{ k!}", "(d^{(k)}_{\\mathbf{X}_0}f)(\\mathbf{U})", "\\end{equation}", "for any constant vector $\\mathbf{U}$.", "To prove \\part{a}, suppose that", "$d^{(k)}_{\\mathbf{X}_0}f$ is not semidefinite. Then there are vectors $\\mathbf{U}_1$ and", "$\\mathbf{U}_2$ such that", "$$", "(d^{(k)}_{\\mathbf{X}_0} f)(\\mathbf{U}_1)>0\\mbox{\\quad and\\quad} (d^{(k)}_\\mathbf{X_0}f)(\\mathbf{U}_2)<0.", "$$", "This and \\eqref{eq:5.4.40} imply that", "$$", "f(\\mathbf{X}_0+t\\mathbf{U}_1)>f(\\mathbf{X}_0)\\mbox{\\quad and\\quad}", " f(\\mathbf{X}_0+t\\mathbf{U}_2)0$ such that", "\\begin{equation} \\label{eq:5.4.41}", "\\frac{(d^{(k)}_{\\mathbf{X}_0} f)(\\mathbf{X}-\\mathbf{X}_0)}{ k!}\\ge\\rho", "|\\mathbf{X}-\\mathbf{X}_0|^k", "\\end{equation}", "\\newpage", "\\noindent", "for all $\\mathbf{X}$ (Exercise~\\ref{exer:5.4.19}). From \\eqref{eq:5.4.39}, there", "is a $\\delta>0$ such that", "$$", "\\frac{f(\\mathbf{X})-f(\\mathbf{X}_0)-\\dst\\frac{1}{k!} (d^{(k)}_{\\mathbf{X}_0}f)(\\mathbf{X}-\\mathbf{X}_0)}{ |\\mathbf{X}-\\mathbf{X}_0|^k}>-", "\\frac{\\rho}{2}\\mbox{\\quad if\\quad} |\\mathbf{X}-\\mathbf{X}_0|<\\delta.", "$$", "Therefore,", "$$", "f(\\mathbf{X})-f(\\mathbf{X}_0)>\\frac{1}{ k!}", "(d^{(k)}_{\\mathbf{X}_0})(\\mathbf{X}-\\mathbf{X}_0)-\\frac{\\rho}{2}", "|\\mathbf{X}-\\mathbf{X}_0|^k\\mbox{\\quad if\\quad}", "|\\mathbf{X}-\\mathbf{X}_0|<\\delta.", "$$", "This and \\eqref{eq:5.4.41} imply that", "$$", "f(\\mathbf{X})-f(\\mathbf{X}_0)>\\frac{\\rho}{2}", " |\\mathbf{X}-\\mathbf{X}_0|^k\\mbox{\\quad if\\quad} |\\mathbf{X}-\\mathbf{X}_0| <\\delta,", "$$", "which implies that $\\mathbf{X}_0$ is a local minimum point of $f$. This proves", "half of \\part{b}. We leave the other half to you", "(Exercise~\\ref{exer:5.4.20}).", "To prove \\part{c} merely requires examples; see Exercise~\\ref{exer:5.4.21}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.4.9" ], "ref_ids": [ 166 ] } ], "ref_ids": [ 166 ] }, { "id": 168, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.2", "categories": [], "title": "", "contents": [ " A transformation $\\mathbf{L}: \\R^n \\to \\R^m$", "defined on all of $\\R^n$ is linear if and only if", "\\begin{equation}\\label{eq:6.1.1}", "\\mathbf{L}(\\mathbf{X})=\\left[\\begin{array}{c} a_{11}x_1+a_{12}x_2+", "\\cdots+a_{1n}x_n\\\\a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n\\\\", "\\vdots\\\\a_{m1}x_1+a_{m2}x_2+\\cdots+a_{mn}x_n\\end{array}\\right],", "\\end{equation}", "where the $a_{ij}$'s are constants$.$" ], "refs": [], "proofs": [ { "contents": [ "If can be seen by induction (Exercise~\\ref{exer:6.1.1}) that if", "$\\mathbf{L}$ is linear, then", "\\begin{equation}\\label{eq:6.1.2}", "\\mathbf{L}(a_1\\mathbf{X}_1+a_2\\mathbf{X}_2+\\cdots+a_k\\mathbf{X}_k)=", "a_1\\mathbf{L}(\\mathbf{X}_1)+a_2\\mathbf{L}(\\mathbf{X}_2)+\\cdots+a_k\\mathbf{L}(\\mathbf{X}_k)", "\\end{equation}", "for any vectors $\\mathbf{X}_1$, $\\mathbf{X}_2$, \\dots, $\\mathbf{X}_k$ and real", "numbers", "$a_1$, $a_2$, \\dots, $a_k$. Any $\\mathbf{X}$ in $\\R^n$ can be", "written as", "\\begin{eqnarray*}", "\\mathbf{X}\\ar=\\left[\\begin{array}{c} x_1\\\\ x_2\\\\\\vdots\\\\ x_n\\end{array}\\right]", "=x_1\\left[\\begin{array}{c} 1\\\\ 0\\\\\\vdots\\\\ 0\\end{array}\\right]", "+x_2\\left[\\begin{array}{c} 0\\\\ 1\\\\\\vdots\\\\ 0\\end{array}\\right]+\\cdots", "+x_n\\left[\\begin{array}{c} 0\\\\ 0\\\\\\vdots\\\\ 1\\end{array}\\right]\\\\", "\\ar=x_1\\mathbf{E}_1+x_2\\mathbf{E}_2+\\cdots+x_n\\mathbf{E}_n.", "\\end{eqnarray*}", "Applying \\eqref{eq:6.1.2} with $k=n$, $\\mathbf{X}_i=\\mathbf{E}_i$, and", "$a_i=x_i$ yields", "\\begin{equation}\\label{eq:6.1.3}", "\\mathbf{L}(\\mathbf{X})=x_1\\mathbf{L}(\\mathbf{E}_1)+x_2\\mathbf{L}(\\mathbf{E}_2)", "+\\cdots+x_n\\mathbf{L}(\\mathbf{E}_n).", "\\end{equation}", "Now denote", "$$", "\\mathbf{L}(\\mathbf{E}_j)=\\left[\\begin{array}{c} a_{1j}\\\\ a_{2j}\\\\", "\\vdots\\\\ a_{mj}\\end{array}\\right],", "$$", "so \\eqref{eq:6.1.3} becomes", "$$", "\\mathbf{L}(\\mathbf{X})=x_1\\left[\\begin{array}{c} a_{11}\\\\ a_{21}\\\\\\vdots\\\\ a_{m1}", "\\end{array}\\right]", "+x_2\\left[\\begin{array}{c} a_{12}\\\\ a_{22}\\\\\\vdots\\\\ a_{m2}\\end{array}", "\\right]+\\cdots", "+x_n\\left[\\begin{array}{c} a_{1n}\\\\ a_{2n}\\\\\\vdots\\\\ a_{mn}\\end{array}", "\\right],", "$$", "which is equivalent to \\eqref{eq:6.1.1}. This proves that if $\\mathbf{L}$ is", "linear, then $\\mathbf{L}$ has the form \\eqref{eq:6.1.1}. We leave the proof of the", "converse to you (Exercise~\\ref{exer:6.1.2})." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 169, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.4", "categories": [], "title": "", "contents": [ " If $\\mathbf{A},$ $\\mathbf{B},$ and $\\mathbf{C}$ are", "$m\\times n$ matrices$,$ then", "$$", "(\\mathbf{A}+\\mathbf{B})+\\mathbf{C}=\\mathbf{A}+(\\mathbf{B}", "+\\mathbf{C}).", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 170, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.5", "categories": [], "title": "", "contents": [ "If $\\mathbf{A}$ and $\\mathbf{B}$ are $m\\times n$", "matrices and $r$ and $s$ are real numbers$,$ then \\part{a}", "$r(s\\mathbf{A})", "=(rs)\\mathbf{A};$ \\part{b} $(r+s)\\mathbf{A}=r\\mathbf{A}+s\\mathbf{A};$", "\\part{c} $r(\\mathbf{A}+\\mathbf{B})=r\\mathbf{A}+r\\mathbf{B}.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 171, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.6", "categories": [], "title": "", "contents": [ " If $\\mathbf{A},$ $\\mathbf{B},$ and $\\mathbf{C}$ are", "$m\\times p,$ $p\\times q,$ and $q\\times n$ matrices$,$ respectively$,$", "then", "$(\\mathbf{AB})\\mathbf{C}=\\mathbf{A}(\\mathbf{BC}).$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 172, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.7", "categories": [], "title": "", "contents": [ "\\begin{alist}", "\\item % (a)", "If we regard the vector", "$$", "\\mathbf{X}=\\left[\\begin{array}{c} x_1\\\\ x_2\\\\\\vdots\\\\", "x_n\\end{array}\\right]", "$$", "as an $n\\times 1$ matrix$,$ then the linear transformation", "$\\eqref{eq:6.1.1}$ can be written as", "$$", "\\mathbf{L}(\\mathbf{X})=\\mathbf{AX}.", "$$", "\\newpage", "\\noindent", "\\item % (b)", "If $\\mathbf{L}_1$ and $\\mathbf{L}_2$ are linear transformations from", "$\\R^n$ to $\\R^m$ with matrices $\\mathbf{A}_1$ and $\\mathbf{A}_{2}$", "respectively$,$ then $c_1\\mathbf{L}_1+c_2\\mathbf{L}_2$ is the linear", "transformation", "from $\\R^n$ to $\\R^m$ with matrix $c_1\\mathbf{A}_1+c_2\\mathbf{A}_{2}.$", "\\item % (c)", "If $\\mathbf{L}_1: \\R^n\\to \\R^p$ and $\\mathbf{L}_2: \\R^p\\to", "\\R^m$ are linear transformations with matrices $\\mathbf{A}_1$ and", "$\\mathbf{A}_2,$ respectively$,$ then the composite function", "$\\mathbf{L}_3=\\mathbf{L}_2\\circ\\mathbf{L}_1,$ defined by", "$$", "\\mathbf{L}_3(\\mathbf{X})=\\mathbf{L}_2(\\mathbf{L}_1(\\mathbf{X})),", "$$", "is the linear transformation from $\\R^n$ to $\\R^m$ with", "matrix $\\mathbf{A}_2\\mathbf{A}_1.$", "\\end{alist}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 173, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.9", "categories": [], "title": "", "contents": [ "If $\\mathbf{A}$ and $\\mathbf{B}$ are $n\\times n$ matrices$,$ then", "$$", "\\det(\\mathbf{A}\\mathbf{B})=\\det(\\mathbf{A})\\det(\\mathbf{B}).", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 174, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.11", "categories": [], "title": "", "contents": [ "Let $\\mathbf{A}$ be an $n\\times n$ matrix$.$", "\\begin{alist}", "\\item % (a)", "The sum of the products of the entries of a row of $\\mathbf{A}$", "and their cofactors equals $\\det(\\mathbf{A}),$ while the", " sum of the products of the entries of a row of $\\mathbf{A}$", "and the cofactors of the entries of a different row equals zero$;$", "that is$,$", "\\begin{equation} \\label{eq:6.1.8}", "\\sum^n_{k=1} a_{ik}c_{jk}=\\left\\{\\casespace\\begin{array}{ll}\\det(\\mathbf{A}),&i=j,\\\\", " 0,&i\\ne j.\\end{array}\\right.", "\\end{equation}", "\\item % (b)", "The sum of the products of the entries of a column of $\\mathbf{A}$", "and their cofactors equals $\\det(\\mathbf{A}),$ while the", " sum of the products of the entries of a column of $\\mathbf{A}$", "and the cofactors of the entries of a different column equals zero$;$", "that is$,$", "\\begin{equation} \\label{eq:6.1.9}", "\\sum^n_{k=1} c_{ki}a_{kj}=\\left\\{\\casespace\\begin{array}{ll}", "\\det(\\mathbf{A}),", "&i=j,\\\\", " 0,&i\\ne j.\\end{array}\\right.", "\\end{equation}", "\\end{alist}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 175, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.12", "categories": [], "title": "", "contents": [ "Let $\\mathbf{A}$ be an $n\\times n$ matrix$.$", "If $\\det(\\mathbf{A})=0,$ then $\\mathbf{A}$ is singular$.$ If", "$\\det(\\mathbf{A})\\ne0,$ then $\\mathbf{A}$ is nonsingular$,$ and $\\mathbf{A}$", "has the unique inverse", "\\begin{equation} \\label{eq:6.1.10}", "\\mathbf{A}^{-1}=\\frac{1}{\\det(\\mathbf{A})}\\adj(\\mathbf{A}).", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "If $\\det(\\mathbf{A})=0$, then $\\det(\\mathbf{A}\\mathbf{B})=0$ for any $n\\times", "n$ matrix, by Theorem~\\ref{thmtype:6.1.9}. Therefore, since", "$\\det(\\mathbf{I})=1$,", " there is no matrix $n\\times n$ matrix $\\mathbf{B}$ such that", "$\\mathbf{A}\\mathbf{B}=\\mathbf{I}$; that is, $\\mathbf{A}$ is singular if", " $\\det(\\mathbf{A})=0$.", " Now suppose that $\\det(\\mathbf{A})\\ne0$. Since \\eqref{eq:6.1.8} implies", "that", "$$", " \\mathbf{A}\\adj(\\mathbf{A})=\\det(\\mathbf{A})\\mathbf{I}", "$$", "and \\eqref{eq:6.1.9} implies that", "$$", " \\adj(\\mathbf{A})\\mathbf{A}=\\det(\\mathbf{A})\\mathbf{I},", "$$", "dividing both sides of these two equations by $\\det(\\mathbf{A})$", "shows that", " if $\\mathbf{A}^{-1}$ is as defined in \\eqref{eq:6.1.10},", "then $\\mathbf{A}\\mathbf{A}^{-1}=\\mathbf{A}^{-1}\\mathbf{A}=\\mathbf{I}$. Therefore,", "$\\mathbf{A}^{-1}$ is an inverse of $\\mathbf{A}$. To see that it is the only", "inverse, suppose that $\\mathbf{B}$ is an $n\\times n$ matrix such that", "$\\mathbf{A}\\mathbf{B}=\\mathbf{I}$. Then", " $\\mathbf{A}^{-1}(\\mathbf{A}\\mathbf{B})=\\mathbf{A}^{-1}$,", " so $(\\mathbf{A}^{-1}\\mathbf{A})\\mathbf{B}=\\mathbf{A}^{-1}$. Since", "$\\mathbf{A}\\mathbf{A}^{-1}=\\mathbf{I}$ and $\\mathbf{I}\\mathbf{B}=\\mathbf{B}$, it follows", "that $\\mathbf{B}=\\mathbf{A}^{-1}$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:6.1.9" ], "ref_ids": [ 173 ] } ], "ref_ids": [] }, { "id": 176, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.13", "categories": [], "title": "", "contents": [ "The system $\\eqref{eq:6.1.11}$ has a solution $\\mathbf{X}$ for any given", "$\\mathbf{Y}$ if and only if $\\mathbf{A}$ is nonsingular$.$ In this case$,$", "the", "solution is unique and is given by $\\mathbf{X}=\\mathbf{A}^{-1}\\mathbf{Y}$." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\mathbf{A}$ is nonsingular, and let", "$\\mathbf{X}=\\mathbf{A}^{-1}\\mathbf{Y}$. Then", "$$", "\\mathbf{A}\\mathbf{X}=\\mathbf{A}(\\mathbf{A}^{-1}\\mathbf{Y})=", "(\\mathbf{A}\\mathbf{A}^{-1})\\mathbf{Y}", "=\\mathbf{I}\\mathbf{Y}=\\mathbf{Y};", "$$", "that is, $\\mathbf{X}$ is a solution of \\eqref{eq:6.1.11}.", "To see that $\\mathbf{X}$ is the only solution of \\eqref{eq:6.1.11},", "suppose that $\\mathbf{A}\\mathbf{X}_1=\\mathbf{Y}$.", " Then $\\mathbf{A}\\mathbf{X}_1=\\mathbf{A}", "\\mathbf{X}$, so", "\\begin{eqnarray*}", "\\mathbf{A}^{-1}(\\mathbf{A}\\mathbf{X})\\ar=", "\\mathbf{A}^{-1}(\\mathbf{A}\\mathbf{X}_1)\\\\", "\\arraytext{and}\\\\", "(\\mathbf{A}^{-1}\\mathbf{A})\\mathbf{X}\\ar=", "(\\mathbf{A}^{-1}\\mathbf{A})\\mathbf{X}_1,", "\\end{eqnarray*}", "which is equivalent to $\\mathbf{I}\\mathbf{X}=\\mathbf{I}\\mathbf{X}_1$, or", "$\\mathbf{X}=\\mathbf{X}_1$.", "Conversely, suppose that \\eqref{eq:6.1.11} has a solution for every", "$\\mathbf{Y}$, and let", " $\\mathbf{X}_i$", "satisfy $\\mathbf{A}\\mathbf{X}_i=\\mathbf{E}_i$, $1\\le i\\le n$. Let", "$$", "\\mathbf{B}=", "[\\mathbf{X}_1\\,\\mathbf{X}_2\\,\\cdots\\,\\mathbf{X}_n];", "$$", "that is, $\\mathbf{X}_1$, $\\mathbf{X}_2$, \\dots, $\\mathbf{X}_n$ are the columns", "of $\\mathbf{B}$. Then", "$$", "\\mathbf{A}\\mathbf{B}=", "[\\mathbf{A}\\mathbf{X}_1\\,\\mathbf{A}\\mathbf{X}_2\\,\\cdots\\,\\mathbf{A}\\mathbf{X}_n]=", "[\\mathbf{E}_1\\,\\mathbf{E}_2\\,\\cdots\\,\\mathbf{E}_n]", "=\\mathbf{I}.", "$$", "To show that $\\mathbf{B}=\\mathbf{A}^{-1}$, we must still show", "that $\\mathbf{B}\\mathbf{A}=\\mathbf{I}$. We first note that,", "since $\\mathbf{A}\\mathbf{B}", "=\\mathbf{I}$ and $\\det(\\mathbf{B}\\mathbf{A})=\\det(\\mathbf{A}\\mathbf{B})=1$", "(Theorem~\\ref{thmtype:6.1.9}), $\\mathbf{B}\\mathbf{A}$ is nonsingular", "(Theorem~\\ref{thmtype:6.1.12}). Now note that", "$$", "(\\mathbf{B}\\mathbf{A})(\\mathbf{B}\\mathbf{A})=", "\\mathbf{B}(\\mathbf{A}\\mathbf{B})\\mathbf{A})=\\mathbf{B}\\mathbf{I}\\mathbf{A};", "$$", "that is,", "$$", "(\\mathbf{B}\\mathbf{A})(\\mathbf{B}\\mathbf{A})=(\\mathbf{B}\\mathbf{A}).", "$$", "Multiplying both sides of this equation on the left by", "$\\mathbf{B}\\mathbf{A})^{-1}$ yields $\\mathbf{B}\\mathbf{A}=\\mathbf{I}$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:6.1.9", "TRENCH_REAL_ANALYSIS-thmtype:6.1.12" ], "ref_ids": [ 173, 175 ] } ], "ref_ids": [] }, { "id": 177, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.14", "categories": [], "title": "", "contents": [ "If $\\mathbf{A}=[a_{ij}]$ is nonsingular$,$ then the solution of", " the system", "\\begin{eqnarray*}", "a_{11}x_1+a_{12}x_2+\\cdots+a_{1n}x_n\\ar=y_1\\\\", "a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n\\ar=y_2\\\\", "&\\vdots& \\\\", "a_{n1}x_1+a_{n2}x_2+\\cdots+a_{nn}x_n\\ar=y_n", "\\end{eqnarray*}", "$($or$,$ in matrix form$,$ $\\mathbf{AX}=\\mathbf{Y}$$)$ is given", "by", "$$", "x_i=\\frac{D_i}{\\det(\\mathbf{A})},\\quad 1\\le i\\le n,", "$$", "where $D_i$ is the determinant of the matrix obtained by replacing the", "$i$th column of $\\mathbf{A}$ with $\\mathbf{Y};$ thus$,$", "$$", "D_1=\\left|\\begin{array}{cccc} y_1&a_{12}&\\cdots&a_{1n}\\\\", "y_2&a_{22}&\\dots&a_{2n}\\\\", "\\vdots&\\vdots&\\ddots&\\vdots\\\\", "y_n&a_{n2}&\\cdots&a_{nn}\\end{array}\\right|,\\quad", "D_2=\\left|\\begin{array}{ccccc} a_{11}&y_1&a_{13}&\\cdots&a_{1n}\\\\", "a_{21}&y_2&a_{23}&\\cdots&a_{2n}\\\\", "\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\", "a_{n1}&y_n&a_{n3}&\\cdots&a_{nn}\\end{array}\\right|,\\quad\\cdots,", "$$", "$$", "D_n=\\left|\\begin{array}{cccc} a_{11}&\\cdots&a_{1,n-1}&y_1\\\\", "a_{21}&\\cdots&a_{2,n-1}&y_2\\\\", "\\vdots&\\vdots&\\ddots&\\vdots\\\\", "a_{n1}&\\cdots&a_{n,n-1}&y_n\\end{array}\\right|.", "$$" ], "refs": [], "proofs": [ { "contents": [ "From Theorems~\\ref{thmtype:6.1.12} and \\ref{thmtype:6.1.13}, the solution of", "$\\mathbf{A}\\mathbf{X}=\\mathbf{Y}$ is", "\\begin{eqnarray*}", "\\left[\\begin{array}{c}", "x_1\\\\x_2\\\\\\vdots\\\\x_n", "\\end{array}\\right]", "=\\mathbf{A}^{-1}\\mathbf{Y}", "\\ar=\\frac{1}{\\det(\\mathbf{A})}", "\\left[\\begin{array}{cccc}", "c_{11}&c_{21}&\\cdots&c_{n1}\\\\", "c_{12}&c_{22}&\\cdots&c_{n2}\\\\", "\\cdots&\\cdots&\\ddots&\\cdots\\\\", "c_{1n}&c_{2n}&\\cdots&c_{nn}", "\\end{array}\\right]", "\\left[\\begin{array}{c}", "y_1\\\\y_2\\\\\\vdots\\\\y_n", "\\end{array}\\right]\\\\", "\\ar=", "\\left[\\begin{array}{c}", "c_{11}y_1+c_{21}y_2+\\cdots+c_{n1}y_n\\\\", "c_{12}y_1+c_{22}y_2+\\cdots+c_{n2}y_n\\\\", "\\vdots\\\\", "c_{1n}y_1+c_{2n}y_2+\\cdots+c_{nn}y_n", "\\end{array}\\right].", "\\end{eqnarray*}", "But", "$$", "c_{11}y_1+c_{21}y_2+\\cdots+c_{n1}y_n=", "\\left|\\begin{array}{cccc} y_1&a_{12}&\\cdots&a_{1n}\\\\", "y_2&a_{22}&\\dots&a_{2n}\\\\", "\\vdots&\\vdots&\\ddots&\\vdots\\\\", "y_n&a_{n2}&\\cdots&a_{nn}\\end{array}\\right|,", "$$", "\\newpage", "\\noindent", "as can be seen by expanding the determinant on the right", "in cofactors of its first column. Similarly,", "$$", "c_{12}y_1+c_{22}y_2+\\cdots+c_{n2}y_n=", "\\left|\\begin{array}{ccccc} a_{11}&y_1&a_{13}&\\cdots&a_{1n}\\\\", "a_{21}&y_2&a_{23}&\\cdots&a_{2n}\\\\", "\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\", "a_{n1}&y_n&a_{n3}&\\cdots&a_{nn}\\end{array}\\right|,", "$$", "as can be seen by expanding the determinant on the right", "in cofactors of its second column. Continuing in this way completes", "the proof." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:6.1.12", "TRENCH_REAL_ANALYSIS-thmtype:6.1.13" ], "ref_ids": [ 175, 176 ] } ], "ref_ids": [] }, { "id": 178, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.15", "categories": [], "title": "", "contents": [ "The homogeneous system $\\eqref{eq:6.1.12}$ of $n$ equations in $n$", "unknowns has a nontrivial solution if and only if $\\det(\\mathbf{A})=0.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 179, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.16", "categories": [], "title": "", "contents": [ "If $A_1,$ $A_2,$ \\dots$,$ $A_k$ are nonsingular $n\\times n$", "matrices$,$ then so is $A_1A_2\\cdots A_k,$ and", "$$", "(A_1A_2\\cdots A_k)^{-1}=A_k^{-1}A_{k-1}^{-1}\\cdots A_1^{-1}.", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 180, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.1", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{X}_0$ is in$,$ and a limit point of$,$ the domain", "of", "$\\mathbf{F}: \\R^n\\to\\R^m.$ Then $\\mathbf{F}$ is continuous at", "$\\mathbf{X}_0$ if and only if for each $\\epsilon>0$ there is a $\\delta>0$", "such that", "\\begin{equation}\\label{eq:6.2.1}", "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)|<\\epsilon", "\\mbox{\\quad if \\quad} |\\mathbf{X}-\\mathbf{X}_0|<\\delta", "\\mbox{\\quad and \\quad} \\mathbf{X}\\in D_\\mathbf{F}.", "\\end{equation}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 181, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.2", "categories": [], "title": "", "contents": [ "A transformation", "$\\mathbf{F}=(f_1,f_2, \\dots,f_m)$ defined in a neighborhood of", "$\\mathbf{X}_0\\in\\R^n$", " is differentiable at $\\mathbf{X}_0$ if and only if", "there is a constant $m\\times n$ matrix $\\mathbf{A}$ such that", "\\begin{equation}\\label{eq:6.2.2}", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}", "\\frac{", "\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)-\\mathbf{A} (\\mathbf{X}-\\mathbf{X}_0)}", "{|\\mathbf{X}-\\mathbf{X}_0|}=\\mathbf{0}.", " \\end{equation}", "If $\\eqref{eq:6.2.2}$ holds$,$ then $\\mathbf{A}$ is given uniquely by", "\\begin{equation}\\label{eq:6.2.3}", "\\mathbf{A}=\\left[\\frac{\\partial f_i(\\mathbf{X}_0)}{\\partial x_j}\\right]=", "\\left[\\begin{array}{cccc}\\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial", "x_1}}&", "\\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial x_2}}&\\cdots&", "\\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial x_n}}\\\\", "[3\\jot]", "\\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_1}}&", "\\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_2}}&", "\\cdots&\\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_n}}\\\\", "\\vdots&\\vdots&\\ddots&\\vdots\\\\", "\\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x_1}}&", "\\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x _2}}&", "\\cdots&\\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x_n}}", "\\end{array}\\right].", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathbf{X}_0=(x_{10},x_{20}, \\dots,x_{n0})$.", " If $\\mathbf{F}$ is differentiable at $\\mathbf{X}_0$, then so are", "$f_1$, $f_2$, \\dots, $f_m$ (Definition~\\ref{thmtype:5.4.1}).", "Hence,", "$$", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} \\frac{\\dst{f_i(\\mathbf{X})-f_i(\\mathbf{X}_0) -", "\\sum_{j=1}^n \\frac{\\partial", "f_i(\\mathbf{X}_0)}{\\partial x_j} (x_j-x_{j0})}}", "{ |\\mathbf{X}-\\mathbf{X}_{0}|}=0,", "\\quad 1\\le i\\le m,", "$$", "which implies \\eqref{eq:6.2.2} with $\\mathbf{A}$ as in", "\\eqref{eq:6.2.3}.", "Now suppose that \\eqref{eq:6.2.2} holds", "with $\\mathbf{A}=[a_{ij}]$. Since", "each component of the vector in \\eqref{eq:6.2.2}", " approaches zero as $\\mathbf{X}$", " approaches $\\mathbf{X}_0$, it follows that", "$$", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}", " \\frac{\\dst{f_i(\\mathbf{X})-f_i(\\mathbf{X}_0)", "-\\dst{\\sum_{j=1}^n} a_{ij}", "(x_j-x_{j0})}}{ |\\mathbf{X}-\\mathbf{X}_0|}", "=0,\\quad 1\\le i\\le m,", "$$", "so each $f_i$ is differentiable at $\\mathbf{X}_0$, and therefore so", "is $\\mathbf{F}$ (Definition~\\ref{thmtype:5.4.1}).", "By Theorem~\\ref{thmtype:5.3.6},", "$$", "a_{ij}=\\frac{\\partial f_i (\\mathbf{X}_0)}{\\partial x_j},\\quad 1\\le i\\le m,", "\\quad 1\\le j\\le n,", "$$", "which implies \\eqref{eq:6.2.3}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.4.1", "TRENCH_REAL_ANALYSIS-thmtype:5.4.1", "TRENCH_REAL_ANALYSIS-thmtype:5.3.6" ], "ref_ids": [ 351, 351, 158 ] } ], "ref_ids": [] }, { "id": 182, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.3", "categories": [], "title": "", "contents": [ "If $\\mathbf{F}: \\R^n\\to\\R^m$ is differentiable at", "$\\mathbf{X}_0,$ then $\\mathbf{F}$ is continuous at~$\\mathbf{X}_0.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 183, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.4", "categories": [], "title": "", "contents": [ "Let $\\mathbf{F}=(f_1,f_2, \\dots,f_m):\\R^n\\to\\R^m,$ and", "suppose that the partial derivatives", "\\begin{equation}\\label{eq:6.2.7}", "\\frac{\\partial f_i}{\\partial x_j},\\quad 1\\le i\\le m,\\quad 1\\le j\\le", "n,", "\\end{equation}", "exist on a neighborhood of $\\mathbf{X}_0$ and", "are continuous at $\\mathbf{X}_0.$ Then $\\mathbf{F}$ is differentiable at", "$\\mathbf{X}_0.$" ], "refs": [], "proofs": [ { "contents": [ "Consider the auxiliary function", "\\begin{equation} \\label{eq:6.2.9}", "\\mathbf{G}(\\mathbf{X})=\\mathbf{F}(\\mathbf{X})-\\mathbf{F}'(\\mathbf{X}_0)\\mathbf{X}.", "\\end{equation}", "The components of $\\mathbf{G}$ are", "$$", "g_i(\\mathbf{X})=f_i(\\mathbf{X})-\\sum_{j=1}^n", "\\frac{\\partial f_i(\\mathbf{X}_{0})", "\\partial x_j} x_j,", "$$", "so", "$$", "\\frac{\\partial g_i(\\mathbf{X})}{\\partial x_j}=", "\\frac{\\partial f_i(\\mathbf{X})}", "{\\partial x_j}-\\frac{\\partial f_i(\\mathbf{X}_0)}{\\partial x_j}.", "$$", "\\newpage", "\\noindent", "Thus, $\\partial g_i/\\partial x_j$ is continuous on $N$ and zero at", "$\\mathbf{X}_0$. Therefore, there is a $\\delta>0$ such that", "\\begin{equation}\\label{eq:6.2.10}", "\\left|\\frac{\\partial g_i(\\mathbf{X})}{\\partial x_j}\\right|<\\frac{\\epsilon}{", "\\sqrt{mn}}\\mbox{\\quad for \\quad}1\\le i\\le m,\\quad 1\\le j\\le n,", "\\mbox{\\quad if \\quad}", "|\\mathbf{X}-\\mathbf{X}_0|<\\delta.", "\\end{equation}", "Now suppose that $\\mathbf{X}$, $\\mathbf{Y}\\in B_\\delta(\\mathbf{X}_0)$. By", "Theorem~\\ref{thmtype:5.4.5},", "\\begin{equation}\\label{eq:6.2.11}", "g_i(\\mathbf{X})-g_i(\\mathbf{Y})=\\sum_{j=1}^n", "\\frac{\\partial g_i(\\mathbf{X}_i)}{\\partial x_j}(x_j-y_j),", "\\end{equation}", "where $\\mathbf{X}_i$ is on the line segment from $\\mathbf{X}$ to $\\mathbf{Y}$,", "so $\\mathbf{X}_i\\in B_\\delta(\\mathbf{X}_0)$. From \\eqref{eq:6.2.10},", "\\eqref{eq:6.2.11}, and Schwarz's inequality,", "$$", "(g_i(\\mathbf{X})-g_i(\\mathbf{Y}))^2\\le\\left(\\sum_{j=1}^n\\left[\\frac{\\partial", "g_i", "(\\mathbf{X}_i)}{\\partial x_j}\\right]^2\\right)", "|\\mathbf{X}-\\mathbf{Y}|^2", "<\\frac{\\epsilon^2}{ m} |\\mathbf{X}-\\mathbf{Y}|^2.", "$$", "Summing this from $i=1$ to $i=m$ and taking square roots yields", "\\begin{equation}\\label{eq:6.2.12}", "|\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})|<\\epsilon", "|\\mathbf{X}-\\mathbf{Y}|", "\\mbox{\\quad if\\quad}\\mathbf{X}, \\mathbf{Y}\\in B_\\delta(\\mathbf{X}_0).", "\\end{equation}", "To complete the proof, we note that", "\\begin{equation}\\label{eq:6.2.13}", "\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})=", "\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})+\\mathbf{F}'(\\mathbf{X}_0)(\\mathbf{X}-\\mathbf{Y}),", "\\end{equation}", " so \\eqref{eq:6.2.12} and the triangle inequality imply \\eqref{eq:6.2.8}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.4.5" ], "ref_ids": [ 164 ] } ], "ref_ids": [] }, { "id": 184, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.8", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{F}: \\R^n\\to\\R^m$ is differentiable at", "$\\mathbf{X}_0,$ $\\mathbf{G}:\\R^k\\to\\R^n$ is differentiable at", "$\\mathbf{U}_0,$ and $\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0).$ Then the composite", "function $\\mathbf{H}=\\mathbf{F}\\circ\\mathbf{G}:\\R^k\\to\\R^m,$", "defined by", "$$", "\\mathbf{H}(\\mathbf{U})=\\mathbf{F}(\\mathbf{G}(\\mathbf{U})),", "$$", "is differentiable at $\\mathbf{U}_0.$ Moreover$,$", "\\begin{equation}\\label{eq:6.2.22}", "\\mathbf{H}'(\\mathbf{U}_0)=\\mathbf{F}'(\\mathbf{G}(\\mathbf{U}_0))", "\\mathbf{G}'(\\mathbf{U}_0)", "\\end{equation}", "and", "\\begin{equation}\\label{eq:6.2.23}", "d_{\\mathbf{U}_0}\\mathbf{H}=d_{\\mathbf{X}_0}\\mathbf{F}\\circ d_{\\mathbf{U}_0}\\mathbf{G},", "\\end{equation}", "where $\\circ$ denotes composition$.$" ], "refs": [], "proofs": [ { "contents": [ "The components of $\\mathbf{H}$ are $h_1$, $h_2$, \\dots, $h_m$, where", "$$", "h_i(\\mathbf{U})=f_i(\\mathbf{G}(\\mathbf{U})).", "$$", "Applying Theorem~\\ref{thmtype:5.4.3} to $h_i$ yields", "\\begin{equation}\\label{eq:6.2.24}", "d_{\\mathbf{U}_0}h_i=\\sum_{j=1}^n \\frac{\\partial f_i(\\mathbf{X}_{0})}", "{\\partial x_j} d_{\\mathbf{U}_0}g_j,\\quad 1\\le i\\le m.", "\\end{equation}", "\\newpage", "\\enlargethispage{\\baselineskip}", "\\noindent Since", "$$", "d_{\\mathbf{U}_0}\\mathbf{H}=\\left[\\begin{array}{c}", "d_{\\mathbf{U}_0}h_1\\\\ d_{\\mathbf{U}_0}h_2\\\\", "\\vdots\\\\", "d_{\\mathbf{U}_0} h_m\\end{array}\\right]\\mbox{", "\\quad and\\quad} d_{\\mathbf{U}_0}\\mathbf{G}=", "\\left[\\begin{array}{c} d_{\\mathbf{U}_0}g_1\\\\ d_{\\mathbf{U}_0}g_2\\\\", "\\vdots\\\\ d_{\\mathbf{U}_0}g_n", "\\end{array}\\right],", "$$", "\\vskip5pt", "\\noindent the $m$ equations in \\eqref{eq:6.2.24} can be", "written in matrix form as", "\\begin{equation}\\label{eq:6.2.25}", "d_{\\mathbf{U}_0}\\mathbf{H}=\\mathbf{F}'(\\mathbf{X}_0)d_{\\mathbf{U}_0}\\mathbf{G}=", "\\mathbf{F}'(\\mathbf{G}(\\mathbf{U}_0)) d_{\\mathbf{U}_0}\\mathbf{G}.", "\\end{equation}", "But", "$$", "d_{\\mathbf{U}_0}\\mathbf{G}=\\mathbf{G}'(\\mathbf{U}_0)\\,d\\mathbf{U},", "$$", "where", "$$", "d\\mathbf{U}=\\left[\\begin{array}{c} du_1\\\\ du_2\\\\\\vdots\\\\", "du_k\\end{array}\\right],", "$$", "so \\eqref{eq:6.2.25} can be rewritten as", "$$", "d_{\\mathbf{U}_0}\\mathbf{H}=", "\\mathbf{F}'(\\mathbf{G}(\\mathbf{U}_0))", "\\mathbf{G}'(\\mathbf{U}_0)\\,d\\mathbf{U}.", "$$", "On the other hand,", "$$", "d_{\\mathbf{U}_0}\\mathbf{H}=\\mathbf{H}'(\\mathbf{U}_0)\\,d\\mathbf{U}.", "$$", "Comparing the last two equations yields \\eqref{eq:6.2.22}.", "Since $\\mathbf{G}'(\\mathbf{U}_0)$ is the matrix of $d_{\\mathbf{U}_0}\\mathbf{G}$", "and $\\mathbf{F}'(\\mathbf{G}(\\mathbf{U}_0))=\\mathbf{F}'(\\mathbf{X}_0)$ is the matrix", "of $d_{\\mathbf{X}_0}\\mathbf{F}$, Theorem~\\ref{thmtype:6.1.7}\\part{c}", "and", "\\eqref{eq:6.2.22} imply~\\eqref{eq:6.2.23}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.4.3", "TRENCH_REAL_ANALYSIS-thmtype:6.1.7" ], "ref_ids": [ 163, 172 ] } ], "ref_ids": [] }, { "id": 185, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.3.1", "categories": [], "title": "", "contents": [ "The linear transformation", "$$", "\\mathbf{U}=\\mathbf{L}(\\mathbf{X})=\\mathbf{A}\\mathbf{X}\\quad (\\R^n\\to", "\\R^n)", "$$", "is invertible if and only if $\\mathbf{A}$ is nonsingular$,$ in which case", "$R(\\mathbf{L})= \\R^n$ and", "$$", "\\mathbf{L}^{-1}(\\mathbf{U})=\\mathbf{A}^{-1}\\mathbf{U}.", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 186, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.3.3", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{F}: \\R^n\\to \\R^n$ is regular on an open", "set $S,$ and let $\\mathbf{G}=\\mathbf{F}^{-1}_S.$ Then $\\mathbf{F}(S)$ is", "open$,$", "$\\mathbf{G}$ is continuously differentiable on $\\mathbf{F}(S),$ and", "$$", "\\mathbf{G}'(\\mathbf{U})=(\\mathbf{F}'(\\mathbf{X}))^{-1},", "\\mbox{\\quad where\\quad}\\mathbf{U}=\\mathbf{F}(\\mathbf{X}).", "$$", "Moreover$,$ since $\\mathbf{G}$ is one-to-one on $\\mathbf{F}(S),$", " $\\mathbf{G}$ is regular on $\\mathbf{F}(S).$" ], "refs": [], "proofs": [ { "contents": [ "We first show that if $\\mathbf{X}_{0} \\in S$,", " then a neighborhood of $\\mathbf{F}(\\mathbf{X}_0)$ is in", "$\\mathbf{F}(S)$.", "This implies that $\\mathbf{F}(S)$ is open.", "Since $S$ is open, there is a $\\rho>0$ such that", " $\\overline{B_\\rho(\\mathbf{X}_0)}\\subset S$. Let $B$", "be the boundary of $B_\\rho(\\mathbf{X}_0)$; thus,", "\\begin{equation} \\label{eq:6.3.20}", "B=\\set\\mathbf{X}{|\\mathbf{X}-\\mathbf{X}_0|=\\rho}.", "\\end{equation}", "The function", "$$", "\\sigma(\\mathbf{X})=|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)|", "$$", "is continuous on $S$ and therefore on $B$, which is compact. Hence,", "by Theorem~\\ref{thmtype:5.2.12}, there is a point $\\mathbf{X}_1$", "in $B$ where $\\sigma(\\mathbf{X})$ attains its minimum value, say $m$, on", "$B$. Moreover, $m>0$, since $\\mathbf{X}_1\\ne\\mathbf{X}_0$ and $\\mathbf{F}$ is", "one-to-one on $S$. Therefore,", "\\begin{equation} \\label{eq:6.3.21}", "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)|\\ge m>0\\mbox{\\quad if\\quad}", "|\\mathbf{X}-\\mathbf{X}_0|=\\rho.", "\\end{equation}", "The set", "$$", "\\set{\\mathbf{U}}{|\\mathbf{U}-\\mathbf{F}(\\mathbf{X}_0)|0$", "and an open neighborhood $N$ of $\\mathbf{X}_0$ such that $N\\subset S$ and", "\\begin{equation} \\label{eq:6.3.24}", "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)|\\ge\\lambda |\\mathbf{X}-\\mathbf{X}_0|", "\\mbox{\\quad if\\quad}\\mathbf{X}\\in N.", "\\end{equation}", "(Exercise~\\ref{exer:6.2.18} also implies this.) Since $\\mathbf{F}$", "satisfies the hypotheses of the present theorem on $N$, the first part", "of this proof shows that $\\mathbf{F}(N)$ is an open set containing", "$\\mathbf{U}_0=\\mathbf{F} (\\mathbf{X}_0)$. Therefore, there is a", "$\\delta>0$ such that", "$\\mathbf{X}=\\mathbf{G}(\\mathbf{U})$ is in $N$ if $\\mathbf{U}\\in", "B_\\delta(\\mathbf{U}_{0})$.", " Setting $\\mathbf{X}=\\mathbf{G}(\\mathbf{U})$ and $\\mathbf{X}_0 =", "\\mathbf{G}(\\mathbf{U}_0)$ in \\eqref{eq:6.3.24} yields", "$$", "|\\mathbf{F}(\\mathbf{G}(\\mathbf{U}))-\\mathbf{F}(\\mathbf{G}(\\mathbf{U}_0))", "|\\ge\\lambda", "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|\\mbox{\\quad if \\quad}", "\\mathbf{U}\\in B_\\delta (\\mathbf{U}_0).", "$$", "Since $\\mathbf{F}(\\mathbf{G}(\\mathbf{U}))=\\mathbf{U}$, this can be rewritten as", "\\begin{equation} \\label{eq:6.3.25}", "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|\\le\\frac{1}{\\lambda} |\\mathbf{U}-", "\\mathbf{U}_0|\\mbox{\\quad if\\quad}\\mathbf{U}\\in B_\\delta(\\mathbf{U}_0),", "\\end{equation}", "which means that $\\mathbf{G}$ is continuous at $\\mathbf{U}_0$.", "Since $\\mathbf{U}_0$ is an arbitrary point in $\\mathbf{F}(S)$, it follows", "that $\\mathbf{G}$ is continous on $\\mathbf{F}(S)$.", "We will now show that $\\mathbf{G}$ is differentiable at $\\mathbf{U}_0$.", "Since", "$$", "\\mathbf{G}(\\mathbf{F}(\\mathbf{X}))=\\mathbf{X},\\quad\\mathbf{X}\\in S,", "$$", "the chain rule (Theorem~\\ref{thmtype:6.2.8}) implies that", "{\\it if\\/} $\\mathbf{G}$ is differentiable at $\\mathbf{U}_0$, then", "$$", "\\mathbf{G}'(\\mathbf{U}_0)\\mathbf{F}'(\\mathbf{X}_0)=\\mathbf{I}", "$$", "\\newpage", "\\noindent", "(Example~\\ref{example:6.2.3}).", " Therefore, if", "$\\mathbf{G}$ is differentiable at $\\mathbf{U}_0$, the differential matrix", "of $\\mathbf{G}$", "must be", "$$", "\\mathbf{G}'(\\mathbf{U}_0)=[\\mathbf{F}'(\\mathbf{X}_0)]^{-1},", "$$", "so to show that $\\mathbf{G}$ is differentiable at", "$\\mathbf{U}_0$, we must show that if", "\\begin{equation} \\label{eq:6.3.26}", "\\mathbf{H}(\\mathbf{U})=", "\\frac{\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)-", "[\\mathbf{F}'(\\mathbf{X}", "_0)]^{-1} (\\mathbf{U}-\\mathbf{U}_0)}{ |\\mathbf{U}-\\mathbf{U}_0|}\\quad", "(\\mathbf{U}\\ne\\mathbf{U}_0),", "\\end{equation}", "then", "\\begin{equation} \\label{eq:6.3.27}", "\\lim_{\\mathbf{U}\\to\\mathbf{U}_0}\\mathbf{H}(\\mathbf{U})=\\mathbf{0}.", "\\end{equation}", "Since $\\mathbf{F}$ is one-to-one on $S$ and $\\mathbf{F}", "(\\mathbf{G}(\\mathbf{U}))", "=\\mathbf{U}$, it follows that if $\\mathbf{U}\\ne\\mathbf{U}_0$, then", "$\\mathbf{G}(\\mathbf{U})\\ne\\mathbf{G}(\\mathbf{U}_0)$. Therefore, we can multiply", "the numerator and denominator of \\eqref{eq:6.3.26}", " by $|\\mathbf{G}(\\mathbf{U})", "-\\mathbf{G}(\\mathbf{U}_0)|$ to obtain", "$$", "\\begin{array}{rcl}", "\\mathbf{H}(\\mathbf{U})\\ar=", "\\dst\\frac{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_{0}|}", "{|\\mathbf{U}-\\mathbf{U}_0|}", "\\left(\\frac{\\mathbf{G}(\\mathbf{U})-\\mathbf{G}", "(\\mathbf{U}_0)-", "\\left[\\mathbf{F}'(\\mathbf{X}_{0})", "\\right]^{-1}(\\mathbf{U}-\\mathbf{U}_0)}", "{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}\\right)\\\\\\\\", "\\ar=-\\dst\\frac{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}{", "|\\mathbf{U}-\\mathbf{U}_0|}", "\\left[\\mathbf{F}'(\\mathbf{X}_0)\\right]^{-1}", "\\left(\\frac{\\mathbf{U}-\\mathbf{U}_0-\\mathbf{F}'(\\mathbf{X}_0)", "(\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0))", "}{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}\\right)", "\\end{array}", "$$", " if $0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta$.", "Because of \\eqref{eq:6.3.25}, this implies that", "$$", "|\\mathbf{H}(\\mathbf{U})|\\le\\frac{1}{\\lambda}", "\\|[\\mathbf{F}'(\\mathbf{X}_0)]^{-1}\\|", "\\left|\\frac{\\mathbf{U}-\\mathbf{U}_0-\\mathbf{F}'(\\mathbf{X}_0)", "(\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0))}{", "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}\\right|", "$$", " if $0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta$.", "Now let", "$$", "\\mathbf{H}_1(\\mathbf{U})=\\frac{\\mathbf{U}-\\mathbf{U}_0-\\mathbf{F}'(\\mathbf{X}_0)", "(\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0))}{", "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}", "$$", "To complete the proof of \\eqref{eq:6.3.27}, we must show that", "\\begin{equation} \\label{eq:6.3.28}", "\\lim_{\\mathbf{U}\\to\\mathbf{U}_0}\\mathbf{H}_1(\\mathbf{U})=\\mathbf{0}.", "\\end{equation}", "Since $\\mathbf{F}$ is differentiable at $\\mathbf{X}_0$, we know that if", "$$", "\\mathbf{H}_2(\\mathbf{X})=", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}", "\\frac{\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)-\\mathbf{F}'(\\mathbf{X}_0)", "(\\mathbf{X}-\\mathbf{X}_0)}{", "|\\mathbf{X}-\\mathbf{X}_0|},", "$$", "then", "\\begin{equation} \\label{eq:6.3.29}", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}\\mathbf{H}_2(\\mathbf{X})=\\mathbf{0}.", "\\end{equation}", "Since $\\mathbf{F}(\\mathbf{G}(\\mathbf{U}))=\\mathbf{U}$ and $\\mathbf{X}_0=", "\\mathbf{G}(\\mathbf{U}_0)$,", "$$", "\\mathbf{H}_1(\\mathbf{U})=\\mathbf{H}_2(\\mathbf{G}(\\mathbf{U})).", "$$", "\\newpage", "\\noindent", "Now suppose that $\\epsilon>0$. From \\eqref{eq:6.3.29}, there is a", "$\\delta_1>0$ such that", "\\begin{equation} \\label{eq:6.3.30}", "|\\mathbf{H}_2(\\mathbf{X})|<\\epsilon\\mbox{\\quad if \\quad} 0<", "|\\mathbf{X}-\\mathbf{X}_{0}|", "=|\\mathbf{X}-\\mathbf{G}(\\mathbf{U}_0)|<\\delta_1.", "\\end{equation}", "Since $\\mathbf{G}$ is continuous at $\\mathbf{U}_0$, there is a", "$\\delta_2\\in(0,\\delta)$ such that", "$$", "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|<\\delta_1\\mbox{\\quad if \\quad}", "0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta_2.", "$$", "This and \\eqref{eq:6.3.30} imply", "that", "$$", "|\\mathbf{H}_1(\\mathbf{U})|=|\\mathbf{H}_2(\\mathbf{G}(\\mathbf{U}))|<\\epsilon", "\\mbox{\\quad if \\quad} 0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta_2.", "$$", "Since this implies", "\\eqref{eq:6.3.28}, $\\mathbf{G}$", "is differentiable at $\\mathbf{X}_0$.", "Since $\\mathbf{U}_0$ is an arbitrary member of $\\mathbf{F}(N)$, we", "can now drop the zero subscript and conclude that $\\mathbf{G}$", "is continuous and differentiable on $\\mathbf{F}(N)$, and", "$$", "\\mathbf{G}'(\\mathbf{U})=[\\mathbf{F}'(\\mathbf{X})]^{-1},\\quad\\mathbf{U}\\in\\mathbf{F}(N).", "$$", "To see that $\\mathbf{G}$ is \\emph{continuously differentiable} on", "$\\mathbf{F}(N)$, we observe that by", "Theorem~\\ref{thmtype:6.1.14}, each", "entry of $\\mathbf{G}'(\\mathbf{U})$ (that is, each partial derivative", "$\\partial g_i(\\mathbf{U})/\\partial u_j$, $1\\le i, j\\le n$) can be written", "as the ratio, with nonzero denominator, of determinants with", "entries of the form", "\\begin{equation} \\label{eq:6.3.31}", "\\frac{\\partial f_r(\\mathbf{G}(\\mathbf{U}))}{\\partial x_s}.", "\\end{equation}", "Since $\\partial f_r/\\partial x_s$ is continuous on $N$ and $\\mathbf{G}$", "is continuous on $\\mathbf{F}(N)$, Theorem~\\ref{thmtype:5.2.10}", "implies that \\eqref{eq:6.3.31} is continuous on $\\mathbf{F}(N)$. Since a", "determinant is a continuous function of its entries, it now follows", "that the entries of $\\mathbf{G}'(\\mathbf{U})$ are continuous on", "$\\mathbf{F}(N)$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.2.12", "TRENCH_REAL_ANALYSIS-thmtype:5.2.12", "TRENCH_REAL_ANALYSIS-thmtype:5.3.11", "TRENCH_REAL_ANALYSIS-thmtype:6.1.13", "TRENCH_REAL_ANALYSIS-thmtype:6.2.6", "TRENCH_REAL_ANALYSIS-thmtype:6.2.8", "TRENCH_REAL_ANALYSIS-thmtype:6.1.14", "TRENCH_REAL_ANALYSIS-thmtype:5.2.10" ], "ref_ids": [ 152, 152, 162, 176, 257, 184, 177, 150 ] } ], "ref_ids": [] }, { "id": 187, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.3.4", "categories": [], "title": "The Inverse Function Theorem", "contents": [ "Let $\\mathbf{F}: \\R^n\\to \\R^n$ be continuously", "differentiable on an open set $S,$ and", "suppose that $J\\mathbf{F}(\\mathbf{X})\\ne0$ on $S.$ Then$,$ if $\\mathbf{X}_0\\in S,$", "there is an open neighborhood $N$ of $\\mathbf{X}_0$ on which $\\mathbf{F}$ is", "regular$.$ Moreover$,$ $\\mathbf{F}(N)$ is open and $\\mathbf{G}=", "\\mathbf{F}^{-1}_N$ is continuously differentiable on $\\mathbf{F}(N),$", "with", "$$", "\\mathbf{G}'(\\mathbf{U})=\\left[\\mathbf{F}'(\\mathbf{X})\\right]^{-1}\\quad", "\\mbox{ $($where", "$\\mathbf{U}=\\mathbf{F}(\\mathbf{X})$$)$},\\quad \\mathbf{U}\\in\\mathbf{F}(N).", "$$" ], "refs": [], "proofs": [ { "contents": [ "Lemma~\\ref{thmtype:6.2.6} implies that there is an open neighborhood", "$N$ of $\\mathbf{X}_0$ on which $\\mathbf{F}$ is one-to-one. The rest of the", "conclusions then follow from applying Theorem~\\ref{thmtype:6.3.3}", " to $\\mathbf{F}$", " on $N$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:6.2.6", "TRENCH_REAL_ANALYSIS-thmtype:6.3.3" ], "ref_ids": [ 257, 186 ] } ], "ref_ids": [] }, { "id": 188, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.4.1", "categories": [], "title": "The Implicit Function Theorem", "contents": [ "Suppose that $\\mathbf{F}:\\R^{n+m}\\to \\R^m$ is continuously", "differentiable on an open set $S$ of $\\R^{n+m}$ containing", "$(\\mathbf{X}_0,\\mathbf{U}_0).$ Let $\\mathbf{F}(\\mathbf{X}_0,\\mathbf{U}_0)=\\mathbf{0},$", "and suppose that $\\mathbf{F}_\\mathbf{U}(\\mathbf{X}_0,\\mathbf{U}_0)$ is", "nonsingular$.$ Then there is a neighborhood $M$ of", " $(\\mathbf{X}_0,\\mathbf{U}_{0}),$", " contained in $S,$ on which", " $\\mathbf{F}_\\mathbf{U}(\\mathbf{X},\\mathbf{U})$", " is nonsingular", " and a neighborhood $N$ of $\\mathbf{X}_0$ in", "$\\R^n$ on which a unique continuously differentiable", " transformation", "$\\mathbf{G}:", "\\R^n\\to", "\\R^m$ is defined$,$ such that", "$\\mathbf{G}(\\mathbf{X}_0)=\\mathbf{U}_0$ and", "\\begin{equation} \\label{eq:6.4.6}", "(\\mathbf{ X},\\mathbf{G}(\\mathbf{X}))\\in M\\mbox{\\quad and \\quad}", "\\mathbf{F}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))=0\\mbox{\\quad", " if}\\quad\\mathbf{X}\\in N.", "\\end{equation}", "Moreover$,$", "\\begin{equation} \\label{eq:6.4.7}", "\\mathbf{G}'(\\mathbf{X})=-[\\mathbf{F}_\\mathbf{U}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))]^{-1}", "\\mathbf{F}_\\mathbf{X}(\\mathbf{X},\\mathbf{G}(\\mathbf{X})),\\quad \\mathbf{X}\\in N.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Define $\\boldsymbol{\\Phi}:\\R^{n+m}\\to \\R^{n+m}$ by", "\\begin{equation} \\label{eq:6.4.8}", "\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{U})=\\left[\\begin{array}{c} x_1\\\\", "x_2\\\\\\vdots\\\\ x_n\\\\ f_1(\\mathbf{X},\\mathbf{U})\\\\", "[3\\jot]", "f_2(\\mathbf{X},\\mathbf{U})\\\\\\vdots\\\\ f_m(\\mathbf{X},\\mathbf{U})\\end{array}", "\\right]", "\\end{equation}", "or, in ``horizontal''notation by", "\\begin{equation} \\label{eq:6.4.9}", "\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{U})=(\\mathbf{X},\\mathbf{F}(\\mathbf{X},\\mathbf{U})).", "\\end{equation}", "Then $\\boldsymbol{\\Phi}$ is continuously differentiable on $S$ and, since", "$\\mathbf{F}(\\mathbf{X}_0,\\mathbf{U}_0)=\\mathbf{0}$,", "\\begin{equation} \\label{eq:6.4.10}", "\\boldsymbol{\\Phi}(\\mathbf{X}_0,\\mathbf{U}_0)=(\\mathbf{X}_0,\\mathbf{0}).", "\\end{equation}", "The differential matrix of $\\boldsymbol{\\Phi}$ is", "$$", "\\boldsymbol{\\Phi}'=\\left[\\begin{array}{cccccccc}", "1&0&\\cdots&0&0&0&\\cdots&0\\\\", "[3\\jot]", "0&1&\\cdots&0&0&0&\\cdots&0\\\\", "\\vdots&\\vdots&\\ddots&\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\", "0&0&\\cdots&1&0&0&\\cdots&0\\\\", "[3\\jot]", "\\dst{\\frac{\\partial f_1}{\\partial x_1}}&", "\\dst{\\frac{\\partial f_1}{\\partial x_2}}&\\cdots&", "\\dst{\\frac{\\partial f_1}{\\partial x_n}}&", "\\dst{\\frac{\\partial f_1}{\\partial u_1}}&", "\\dst{\\frac{\\partial f_1}{\\partial u_2}}&\\cdots&", "\\dst{\\frac{\\partial f_1}{\\partial u_m}}\\\\", "[3\\jot]", "\\dst{\\frac{\\partial f_2}{\\partial x_1}}&", "\\dst{\\frac{\\partial f_2}{\\partial x_2}}&\\cdots&", "\\dst{\\frac{\\partial f_2}{\\partial x_n}}&", "\\dst{\\frac{\\partial f_2}{\\partial u_1}}&", "\\dst{\\frac{\\partial f_2}{\\partial u_2}}&\\cdots&", "\\dst{\\frac{\\partial f_2}{\\partial u_m}}\\\\", "[3\\jot]", "\\vdots&\\vdots&\\ddots&\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\", "[3\\jot]", "\\dst{\\frac{\\partial f_m}{\\partial x_1}}&", "\\dst{\\frac{\\partial f_m}{\\partial x_2}}&\\cdots&", "\\dst{\\frac{\\partial f_m}{\\partial x_n}}&", "\\dst{\\frac{\\partial f_m}{\\partial u_1}}&", "\\dst{\\frac{\\partial f_m}{\\partial u_2}}&\\cdots&", "\\dst{\\frac{\\partial f_m}{\\partial u_m}}\\end{array}\\right]=", "\\left[\\begin{array}{cc}\\mathbf{I}&\\mathbf{0}\\\\\\mathbf{F}_\\mathbf{X}&\\mathbf{F}_\\mathbf{U}", "\\end{array}\\right],", "$$", "\\newpage", "\\noindent", "where $\\mathbf{I}$ is the $n\\times n$ identity matrix, $\\mathbf{0}$ is the", "$n\\times m$ matrix with all zero entries, and $\\mathbf{F}_\\mathbf{X}$ and", "$\\mathbf{F}_\\mathbf{U}$ are as in \\eqref{eq:6.4.5}. By expanding", "$\\det(\\boldsymbol{\\Phi}')$ and the determinants that evolve from it in terms", "of the cofactors of their first rows, it can be shown in $n$ steps", "that", "\\vskip.5pc", "$$", "J\\boldsymbol{\\Phi}=\\det(\\boldsymbol{\\Phi}')=\\left|\\begin{array}{cccc}", "\\dst{\\frac{\\partial f_1}{\\partial u_1}}&", "\\dst{\\frac{\\partial f_1}{\\partial u_2}}&\\cdots&", "\\dst{\\frac{\\partial f_1}{\\partial u_m}}\\\\", "[3\\jot]", "\\dst{\\frac{\\partial f_2}{\\partial u_1}}&", "\\dst{\\frac{\\partial f_2}{\\partial u_2}}&\\cdots&", "\\dst{\\frac{\\partial f_2}{\\partial u_m}}\\\\", "[3\\jot]", "\\vdots&\\vdots&\\ddots&\\vdots\\\\", "\\dst{\\frac{\\partial f_m}{\\partial u_1}}&", "\\dst{\\frac{\\partial f_m}{\\partial u_2}}&\\cdots&", "\\dst{\\frac{\\partial f_m}{\\partial u_m}}\\end{array}\\right|=", "\\det(\\mathbf{F}_\\mathbf{U}).", "$$", "\\vskip.5pc", "In particular,", "$$", "J\\boldsymbol{\\Phi}(\\mathbf{X}_0,\\mathbf{U}_0)=\\det(\\mathbf{F}_\\mathbf{U}", "(\\mathbf{X}_0,\\mathbf{U}_{0})\\ne0.", "$$", "Since $\\boldsymbol{\\Phi}$ is continuously differentiable on $S$,", "Corollary~\\ref{thmtype:6.3.5} implies that $\\boldsymbol{\\Phi}$ is regular", "on some open neighborhood $M$ of $(\\mathbf{X}_0,\\mathbf{U}_0)$ and that", "$\\widehat{M}=\\boldsymbol{\\Phi}(M)$ is open.", "Because of the form of $\\boldsymbol{\\Phi}$ (see \\eqref{eq:6.4.8} or", "\\eqref{eq:6.4.9}),", "we can write points of $\\widehat{M}$ as $(\\mathbf{X},\\mathbf{V})$,", " where $\\mathbf{V}\\in \\R^m$.", "Corollary~\\ref{thmtype:6.3.5} also", "implies that $\\boldsymbol{\\Phi}$ has a a continuously differentiable", "inverse $\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{V})$", "defined on $\\widehat{M}$", "with values in $M$. Since $\\boldsymbol{\\Phi}$ leaves the ``$\\mathbf{X}$", "part\"", "of $(\\mathbf{X},\\mathbf{U})$ fixed, a local inverse of $\\boldsymbol{\\Phi}$", "must also have this property.", " Therefore, $\\boldsymbol{\\Gamma}$ must", "have the form", "\\vskip.5pc", "$$", "\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{V})=\\left[\\begin{array}{c} x_1\\\\", "x_2\\\\\\vdots\\\\ x_n\\\\[3\\jot]", "h_1(\\mathbf{X},\\mathbf{V})\\\\[3\\jot] h_2(\\mathbf{X},\\mathbf{V})\\\\", "\\vdots\\\\", "[3\\jot]", "h_m(\\mathbf{X},\\mathbf{V})\\end{array}\\right]", "$$", "\\vskip1pc", "\\noindent or, in ``horizontal'' notation,", "$$", "\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{V})=(\\mathbf{X},\\mathbf{H}(\\mathbf{X},\\mathbf{V})),", "$$", "\\noindent where $\\mathbf{H}:\\R^{n+m}\\to \\R^m$ is continuously", "differentiable on $\\widehat{M}$.", "We will show that", "$\\mathbf{G}(\\mathbf{X})=\\mathbf{H}(\\mathbf{X},\\mathbf{0})$", "has the stated properties.", "\\enlargethispage{.5\\baselineskip}", "From \\eqref{eq:6.4.10}, $(\\mathbf{X}_0,\\mathbf{0})\\in\\widehat{M}$ and, since", "$\\widehat{M}$ is open, there is a neighborhood $N$ of $\\mathbf{X}_0$ in", "$\\R^n$ such that $(\\mathbf{X},\\mathbf{0})\\in\\widehat{M}$ if $\\mathbf{X}\\in", "N$ (Exercise~\\ref{exer:6.4.2}).", "Therefore, $(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))", "=\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{0})\\in M$ if $\\mathbf{X}\\in N$.", " Since $\\boldsymbol{\\Gamma}=\\boldsymbol{\\Phi}^{-1}$,", "$(\\mathbf{X},\\mathbf{0})", "=\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))$. Setting", "$\\mathbf{X}=\\mathbf{X}_0$ and recalling \\eqref{eq:6.4.10}", "shows that $\\mathbf{G}(\\mathbf{X}_0)=\\mathbf{U}_0$, since $\\boldsymbol{\\Phi}$", "is one-to-one on $M$.", "\\newpage", "Henceforth we assume that $\\mathbf{X}\\in N$.", "Now,", "$$", "\\begin{array}{rcll}", "(\\mathbf{X},\\mathbf{0})\\ar=", "\\boldsymbol{\\Phi}(\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{0}))", "&\\mbox{", "(since", "$\\boldsymbol{\\Phi}=\\boldsymbol{\\Gamma}^{-1})$}\\\\", "\\ar=\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))&\\mbox{ (since", "$\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{0})=(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))$)}\\\\", "\\ar=(\\mathbf{X},\\mathbf{F}(\\mathbf{X},\\mathbf{G}(\\mathbf{X})))&\\mbox{ (since", "$\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{U})=", "(\\mathbf{X},\\mathbf{F}(\\mathbf{X},\\mathbf{U} ))$)}.", "\\end{array}", "$$", "Therefore, $\\mathbf{F}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))=\\mathbf{0}$; that is,", "$\\mathbf{G}$ satisfies", "\\eqref{eq:6.4.6}.", "To see that $\\mathbf{G}$ is unique,", "suppose that $\\mathbf{G}_1:\\R^n\\to \\R^m$ also satisfies", "\\eqref{eq:6.4.6}. Then", "$$", "\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))=", "(\\mathbf{X},\\mathbf{F}", "(\\mathbf{X},\\mathbf{G}(\\mathbf{X})))=(\\mathbf{X},\\mathbf{0})", "$$", "and", "$$", "\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{G}_1(\\mathbf{X}))=(\\mathbf{X},\\mathbf{F}", "(\\mathbf{X},\\mathbf{G}_1(\\mathbf{X})))=(\\mathbf{X},\\mathbf{0})", "$$", "for all $\\mathbf{X}$ in $N$.", "Since $\\boldsymbol{\\Phi}$ is one-to-one on $M$,", "this implies that $\\mathbf{G}(\\mathbf{X})=", "\\mathbf{G}_1(\\mathbf{X})$.", "Since the partial derivatives", "$$", "\\frac{\\partial h_i}{\\partial x_j},\\quad 1\\le i\\le m,\\quad 1\\le j\\le", "n,", "$$", "are continuous functions of $(\\mathbf{X},\\mathbf{V})$ on $\\widehat{M}$, they", "are continuous with respect to $\\mathbf{X}$ on the subset", "$\\set{(\\mathbf{X},\\mathbf{0})}{\\mathbf{X} \\in N}$ of $\\widehat{M}$.", "Therefore,", "$\\mathbf{G}$ is", "continuously differentiable on $N$. To verify \\eqref{eq:6.4.7}, we write", "$\\mathbf{F}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))=\\mathbf{0}$ in terms of components;", "thus,", "$$", "f_i(x_1,x_2, \\dots,x_n,g_1(\\mathbf{X}),g_2(\\mathbf{X}), \\dots,g_m(\\mathbf{X}))", "=0,\\quad 1\\le i\\le m,\\quad\\mathbf{X}\\in N.", "$$", "Since $f_i$ and $g_1$, $g_2$, \\dots, $g_m$ are continuously", "differentiable on their respective domains, the chain rule", "(Theorem~\\ref{thmtype:5.4.3}) implies that", "\\begin{equation} \\label{eq:6.4.11}", "\\frac{\\partial f_i(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))}{\\partial x_j}+", "\\sum^m_{r=1}", "\\frac{\\partial f_i(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))}{\\partial u_r}", "\\frac{\\partial g_r(\\mathbf{X})", "}{\\partial x_j}=0,\\quad 1\\le i\\le m,\\ 1\\le j\\le n,", "\\end{equation}", "or, in matrix form,", "\\begin{equation} \\label{eq:6.4.12}", "\\mathbf{F}_\\mathbf{X}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))+\\mathbf{F}_\\mathbf{U}", "(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))\\mathbf{G}'(\\mathbf{X})=\\mathbf{0}.", "\\end{equation}", "Since $(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))\\in M$ for all $\\mathbf{X}$", "in $N$ and $\\mathbf{F}_\\mathbf{U}(\\mathbf{X},\\mathbf{U})$ is nonsingular when", "$(\\mathbf{X},\\mathbf{U})\\in M$, we can multiply \\eqref{eq:6.4.12} on the left by", "$\\mathbf{F}^{-1}_\\mathbf{U}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))$ to obtain", "\\eqref{eq:6.4.7}. This completes the proof." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:6.3.5", "TRENCH_REAL_ANALYSIS-thmtype:6.3.5", "TRENCH_REAL_ANALYSIS-thmtype:5.4.3" ], "ref_ids": [ 293, 293, 163 ] } ], "ref_ids": [] }, { "id": 189, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.3", "categories": [], "title": "", "contents": [ "If $f$ is unbounded on the nondegenerate rectangle $R$ in", "$\\R^n,$ then $f$ is not integrable on $R.$" ], "refs": [], "proofs": [ { "contents": [ "We will show that if $f$ is unbounded on $R$, ${\\bf", "P}=\\{R_1,R_2, \\dots,R_k\\}$ is", "any partition of $R$, and $M>0$, then there are Riemann sums $\\sigma$", "and $\\sigma'$ of $f$ over ${\\bf P}$ such that", "\\begin{equation} \\label{eq:7.1.11}", "|\\sigma-\\sigma'|\\ge M.", "\\end{equation}", "This implies that", "$f$ cannot satisfy Definition~\\ref{thmtype:7.1.2}. (Why?)", "Let", "$$", "\\sigma=\\sum_{j=1}^kf(\\mathbf{X}_j)V(R_j)", "$$", "be a Riemann sum of $f$ over ${\\bf P}$. There must be", "an integer $i$ in $\\{1,2, \\dots,k\\}$ such that", "\\begin{equation} \\label{eq:7.1.12}", "|f(\\mathbf{X})-f(\\mathbf{X}_i)|\\ge\\frac{M }{ V(R_i)}", "\\end{equation}", "for some $\\mathbf{X}$ in $R_i$, because if this were not so, we", "would have", "$$", "|f(\\mathbf{X})-f(\\mathbf{X}_j)|<\\frac{M}{ V(R_j)},\\quad \\mathbf{X}\\in R_j,\\quad", "\\quad 1\\le j\\le k.", "$$", "If this is so, then", "\\begin{eqnarray*}", "|f(\\mathbf{X})|\\ar=|f(\\mathbf{X}_j)+f(\\mathbf{X})-f(\\mathbf{X}_j)|\\le|f(\\mathbf{X}_j)|+|f(\\mathbf{X})-f(\\mathbf{X}_j)|\\\\", "\\ar\\le |f(\\mathbf{X}_j)|+\\frac{M}{ V(R_j)},\\quad \\mathbf{X}\\in R_j,\\quad", "1\\le j\\le k.", "\\end{eqnarray*}", "However, this implies that", "$$", "|f(\\mathbf{X})|\\le\\max\\set{|f(\\mathbf{X}_j)|+\\frac{M}{ V(R_j)}}{1\\le j\\le k},", "\\quad \\mathbf{X}\\in R,", "$$", "which contradicts the assumption that $f$ is unbounded on $R$.", " Now suppose that $\\mathbf{X}$ satisfies \\eqref{eq:7.1.12}, and", "consider the Riemann sum", "$$", "\\sigma'=\\sum_{j=1}^nf(\\mathbf{X}_j')V(R_j)", "$$", "over the same partition ${\\bf P}$, where", "$$", "\\mathbf{X}_j'=\\left\\{\\casespace\\begin{array}{ll}", "\\mathbf{X}_j,&j \\ne i,\\\\", "\\mathbf{X},&j=i.\\end{array}\\right.", "$$", "Since", "$$", "|\\sigma-\\sigma'|=|f(\\mathbf{X})-f(\\mathbf{X}_i)|V(R_i),", "$$", "\\eqref{eq:7.1.12} implies \\eqref{eq:7.1.11}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.1.2" ], "ref_ids": [ 359 ] } ], "ref_ids": [] }, { "id": 190, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.5", "categories": [], "title": "", "contents": [ "Let $f$ be bounded on a rectangle $R$ and let $\\mathbf{P}$", "be a partition of $R.$ Then", "\\begin{alist}", "\\item % (a)", " The upper sum $S(\\mathbf{P})$ of $f$ over $\\mathbf{P}$ is the supremum", "of the set of all Riemann sums of $f$ over $\\mathbf{P}.$", "\\item % (b)", " The lower sum $s(\\mathbf{P})$ of $f$ over $\\mathbf{P}$ is the infimum", " of the set of all Riemann sums of $f$ over $\\mathbf{P}.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.5}.", "If", "$$", "m\\le f(\\mathbf{X})\\le M\\mbox{\\quad for $\\mathbf{X}$ in $R$},", "$$", "then", "$$", "mV(R)\\le s({\\bf P})\\le S({\\bf P})\\le MV(R);", "$$", "therefore, $\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}$ and", "$\\underline{\\int_R}\\, f(\\mathbf{X})\\, d\\mathbf{X}$ exist, are unique, and", "satisfy the inequalities", "$$", "mV(R)\\le\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\le MV(R)", "$$", "and", "$$", "mV(R)\\le\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\le MV(R).", "$$", "The upper and lower integrals are also written as", "$$", "\\overline{\\int_R}\\, f(x,y) \\,d(x,y)\\mbox{\\quad and\\quad}\\underline{\\int_R}\\,", "f(x,y) \\,d(x,y)\\quad (n=2),", "$$", "$$", "\\overline{\\int_R}\\, f(x,y,z) \\,d(x,y,z)\\mbox{\\quad and\\quad}", "\\underline{\\int_R}\\, f(x,y,z) \\,d(x,y,z)\\quad (n=3),", "$$", "or", "$$", "\\overline{\\int_R}\\, f(x_1,x_2, \\dots,x_n) \\,d(x_1,x_2, \\dots,x_n)", "$$", "and", "$$", "\\underline{\\int_R}\\, f(x_1,x_2, \\dots,x_n)\\,d(x_1,x_2, \\dots,x_n)\\quad", "\\mbox{\\quad ($n$ arbitrary)}.", "$$", "\\begin{example}\\label{example:7.1.2}\\rm", "Find $\\underline{\\int_R}\\,f(x,y)\\,d(x,y)$ and", " $\\overline{\\int_R}\\,f(x,y)\\,d(x,y)$, with", "$R=[a,b]\\times [c,d]$ and", "$$", "f(x,y)=x+y,", "$$", "as in Example~\\ref{example:7.1.1}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 191, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.7", "categories": [], "title": "", "contents": [ "If $f$ is bounded on a rectangle $R,$ then", "$$", "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}", "\\le\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.8}.", "The next theorem is analogous to Theorem~3.2.3." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 192, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.8", "categories": [], "title": "", "contents": [ "If $f$ is integrable on a rectangle $R,$ then", "$$", "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=", "\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X} =\\int_R f(\\mathbf{X})\\,d\\mathbf{X}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.9}.", "\\newpage", "\\enlargethispage{\\baselineskip}", "\\begin{lemma}\\label{thmtype:7.1.9}", "If $f$ is bounded on a rectangle $R$ and $\\epsilon>0,$ there is", " a $\\delta>0$ such that", "\\vspace{4pt}", "$$", "\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\le S({\\bf P})<\\overline{\\int_R}\\,", "f(\\mathbf{X})\\,d\\mathbf{X}+\\epsilon", "$$", "\\vspace{4pt}", "and", "\\vspace{4pt}", "$$", "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\ge s({\\bf P})>", "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}-\\epsilon", "$$", "\\vspace{4pt}", "if $\\|{\\bf P}\\|<\\delta.$", "\\end{lemma}" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 193, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.10", "categories": [], "title": "", "contents": [ "If $f$ is bounded on a rectangle $R$ and", "\\vspace{2pt}", "$$", "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=", "\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=L,", "$$", "\\vspace{2pt}", "then $f$ is integrable on $R,$ and", "\\vspace{2pt}", "$$", "\\int_R f(\\mathbf{X})\\,d\\mathbf{X}=L.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.11}.", "Theorems~\\ref{thmtype:7.1.8} and \\ref{thmtype:7.1.10}", " imply the following theorem, which is analogous to", "Theorem~\\ref{thmtype:3.2.6}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.1.8", "TRENCH_REAL_ANALYSIS-thmtype:7.1.10", "TRENCH_REAL_ANALYSIS-thmtype:3.2.6" ], "ref_ids": [ 192, 193, 49 ] } ], "ref_ids": [] }, { "id": 194, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.11", "categories": [], "title": "", "contents": [ "A bounded", "function $f$ is integrable on a rectangle $R$ if and only if", "$$", "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=\\overline{\\int_R}\\, f(\\mathbf{X})\\,", "d\\mathbf{X}.", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 195, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.12", "categories": [], "title": "", "contents": [ "If $f$ is bounded on a rectangle $R,$ then $f$ is integrable on $R$", "if and only if for every $\\epsilon>0$ there is a partition ${\\bf P}$", "of $R$ such that", "$$", "S({\\bf P})-s({\\bf P})<\\epsilon.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.12}.", "Theorem~\\ref{thmtype:7.1.12} provides a useful criterion for", "integrability. The next theorem is an important application.", "It is analogous to", "Theorem~\\ref{thmtype:3.2.8}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.1.12", "TRENCH_REAL_ANALYSIS-thmtype:3.2.8" ], "ref_ids": [ 195, 51 ] } ], "ref_ids": [] }, { "id": 196, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.13", "categories": [], "title": "", "contents": [ "If $f$ is continuous on a rectangle $R$ in $\\R^n,$ then $f$ is", "integrable on~$R.$" ], "refs": [], "proofs": [ { "contents": [ "Let $\\epsilon>0$. Since $f$ is uniformly continuous on $R$", "(Theorem~\\ref{thmtype:5.2.14}), there is a $\\delta>0$ such that", "\\begin{equation} \\label{eq:7.1.23}", "|f(\\mathbf{X})-f(\\mathbf{X}')|<\\frac{\\epsilon}{ V({\\bf R})}", "\\end{equation}", "if $\\mathbf{X}$ and $\\mathbf{X}'$ are in $R$ and", " $|\\mathbf{X}-\\mathbf{X}'|<\\delta$. Let ${\\bf P}=\\{R_1,R_2, \\dots,R_k\\}$ be a partition of", "$R$ with $\\|P\\|<\\delta/\\sqrt n$. Since $f$ is continuous on $R$, there", "are points $\\mathbf{X}_j$ and $\\mathbf{X}_j'$ in $R_j$ such that", "$$", "f(\\mathbf{X}_j)=M_j=\\sup_{\\mathbf{X}\\in R_j}f(\\mathbf{X})", "\\mbox{\\quad and \\quad}", "f(\\mathbf{X}_j')=m_j=\\inf_{\\mathbf{X}\\in R_j}f(\\mathbf{X})", "$$", "(Theorem~\\ref{thmtype:5.2.12}).", "Therefore,", "$$", "S(\\mathbf{P})-s(\\mathbf{P})=\\sum_{j=1}^n(f(\\mathbf{X}_j)-", "f(\\mathbf{X}_j'))V(R_j).", "$$", "Since $\\|{\\bf P}\\|<\\delta/\\sqrt n$,", "$|\\mathbf{X}_j-\\mathbf{X}_j'|<\\delta$, and, from \\eqref{eq:7.1.23}", "with $\\mathbf{X}=\\mathbf{X}_j$ and $\\mathbf{X}'=\\mathbf{X}_j'$,", "$$", " S(\\mathbf{P})-s(\\mathbf{P})<\\frac{\\epsilon}{ V(R)}", "\\sum_{j=1}^kV(R_j)=\\epsilon.", "$$", "Hence, $f$ is integrable", "on $R$, by Theorem~\\ref{thmtype:7.1.12}.", "\\boxit{Sets with Zero Content}", "The next definition will enable us to establish the existence", "of $\\int_Rf(\\mathbf{X})\\,d\\mathbf{X}$ in cases where $f$ is bounded on the", "rectangle $R$, but is not necessarily continuous for all $\\mathbf{X}$", "in $R$.", "\\begin{definition}\\label{thmtype:7.1.14}", "A subset $E$ of $\\R^n$ has zero content if for each", "$\\epsilon>0$", "there is a finite set of rectangles $T_1$, $T_2$, \\dots, $T_m$ such", "that", "\\begin{equation}\\label{eq:7.1.24}", "E\\subset\\bigcup_{j=1}^m T_j", "\\end{equation}", "and", "\\begin{equation}\\label{eq:7.1.25}", "\\sum_{j=1}^m V(T_j)<\\epsilon.", "\\end{equation}", "\\end{definition}", "\\begin{example}\\label{example:7.1.3}\\rm Since the empty set is contained", "in every rectangle, the empty set has zero content. If $E$ consists of", "finitely", "many points $\\mathbf{X}_1$, $\\mathbf{X}_2$, \\dots,", "$\\mathbf{X}_m$, then $\\mathbf{X}_j$ can be enclosed in a rectangle $T_j$", "such that", "$$", "V(T_j)<\\frac{\\epsilon}{ m},\\quad 1\\le j\\le m.", "$$", "Then \\eqref{eq:7.1.24} and \\eqref{eq:7.1.25} hold, so $E$ has zero content.", "\\end{example}", "\\begin{example}\\label{example:7.1.4}\\rm Any bounded set $E$ with only", "finitely many limit points has zero content. To see this, we first", "observe that if $E$ has no limit points, then it must be finite, by", "the Bolzano--Weierstrass theorem (Theorem~\\ref{thmtype:1.3.8}), and", "therefore must have zero content,", "by Example~\\ref{example:7.1.3}. Now suppose that the limit points of $E$ are", "$\\mathbf{X}_1$, $\\mathbf{X}_2$, \\dots, $\\mathbf{X}_m$. Let $R_1$, $R_2$,", "\\dots, $R_m$ be rectangles such that", "$\\mathbf{X}_i\\in R^0_i$ and", "\\begin{equation}\\label{eq:7.1.26}", "V(R_i)<\\frac{\\epsilon}{2m},\\quad 1\\le i\\le m.", "\\end{equation}", "The set of points of $E$ that are not in $\\cup_{j=1}^mR_j$ has no", "limit points (why?) and, being bounded, must be finite (again by the", "Bolzano--Weierstrass theorem). If this set contains $p$ points,", "then it can be covered by rectangles", "$R_1'$, $R_2'$, \\dots, $R_p'$ with", "\\begin{equation}\\label{eq:7.1.27}", "V(R_j')<\\frac{\\epsilon}{2p},\\quad 1\\le j\\le p.", "\\end{equation}", "Now,", "$$", "E\\subset\\left(\\bigcup_{i=1}^mR_i\\right)\\bigcup\\left(\\bigcup^p_{j=1}", "R_j'\\right)", "$$", "and, from \\eqref{eq:7.1.26} and \\eqref{eq:7.1.27},", "$$", "\\sum_{i=1}^m V(R_i)+\\sum_{j=1}^p V(R_j')<\\epsilon.", "$$", "\\end{example}", "\\begin{example}\\label{example:7.1.5}\\rm", " If $f$ is continuous on $[a,b]$,", "then the curve", "\\begin{equation}\\label{eq:7.1.28}", "y=f(x),\\quad a\\le x\\le b", "\\end{equation}", "(that is, the set $\\set{(x,y)}{y=f(x),\\ a\\le x\\le b})$, has zero", "content in $\\R^2$. To see this, suppose that $\\epsilon>0$, and", "choose $\\delta>0$ such that", "\\begin{equation}\\label{eq:7.1.29}", "|f(x)-f(x')|<\\epsilon\\mbox{\\quad if\\quad} x, x'\\in [a,b]", "\\mbox{\\quad and\\quad} |x-x'|<\\delta.", "\\end{equation}", "This is possible because $f$ is uniformly continuous on $[a,b]$", "(Theorem~\\ref{thmtype:2.2.12}). Let", "$$", "P: a=x_00$. Since $E$ has zero content, there are", "rectangles", "$T_1$, $T_2$, \\dots, $T_m$ such that", "\\begin{equation} \\label{eq:7.1.31}", "E\\subset\\bigcup_{j=1}^m T_j", "\\end{equation}", "and", "\\begin{equation} \\label{eq:7.1.32}", "\\sum_{j=1}^m V(T_j)<\\epsilon.", "\\end{equation}", " We may assume that", "$T_1$, $T_2$, \\dots, $T_m$ are contained in $R$, since, if not, their", "intersections with", "$R$ would be contained in $R$, and still satisfy \\eqref{eq:7.1.31}", "and \\eqref{eq:7.1.32}.", " We may also assume that if $T$ is any rectangle such", "that", "\\begin{equation}\\label{eq:7.1.33}", "T\\bigcap\\left(\\bigcup_{j=1}^m T_j^0\\right)=\\emptyset, \\mbox{\\quad", "then", "\\quad}T\\cap E=\\emptyset", "\\end{equation}", "\\newpage", "\\noindent", "since if this were not so, we could make it so by enlarging", "$T_1$, $T_2$, \\dots, $T_m$", "slightly while maintaining \\eqref{eq:7.1.32}. Now suppose that", "\\vspace*{1pt}", "$$", "T_j=[a_{1j},b_{1j}]\\times [a_{2j},b_{2j}]\\times\\cdots\\times", "[a_{nj},b_{nj}],\\quad 1\\le j\\le m,", "$$", "\\vspace*{1pt}", "\\noindent let $P_{i0}$ be the partition of $[a_i,b_i]$ (see", "\\eqref{eq:7.1.30}) with partition points", "$$", "a_i,b_i,a_{i1},b_{i1},a_{i2},b_{i2}, \\dots,a_{im},b_{im}", "\\vspace*{1pt}", "$$", "(these are not in increasing order), $1\\le i\\le n$, and let", "\\vspace*{1pt}", "$$", "{\\bf P}_0=P_{10}\\times P_{20}\\times\\cdots\\times P_{n0}.", "$$", "\\vspace*{1pt}", "\\noindent\\hskip-.3em Then ${\\bf P}_0$ consists of rectangles whose", "union equals $\\cup_{j=1}^m T_j$", "and other rectangles", "$T'_1$, $T'_2$, \\dots, $T'_k$ that do not intersect $E$. (We need", "\\eqref{eq:7.1.33} to be sure that $T'_i\\cap E=\\emptyset,", "1\\le i\\le k.)$ If we let", "$$", "B=\\bigcup_{j=1}^m T_j\\mbox{\\quad and\\quad} C=\\bigcup^k_{i=1} T'_i,", "$$", "then $R=B\\cup C$ and $f$ is continuous on the compact set $C$.", "If ${\\bf P}=\\{R_1,R_2, \\dots,R_k\\}$ is a refinement of ${\\bf P}_0$,", "then every subrectangle $R_j$ of ${\\bf P}$ is contained entirely in", "$B$ or entirely in $C$. Therefore, we can write", "\\vspace*{1pt}", "\\begin{equation}\\label{eq:7.1.34}", "S({\\bf P})-s({\\bf P})=\\Sigma_1(M_j-m_j)", "V(R_j)+\\Sigma_2(M_j-m_j)V(R_j),", "\\end{equation}", "\\vspace*{1pt}", "\\noindent \\hskip-.3em", "where $\\Sigma_1$ and $\\Sigma_2$ are summations over values of $j$ for", "which $R_j\\subset B$ and $R_j\\subset C$, respectively. Now suppose that", "$$", "|f(\\mathbf{X})|\\le M\\mbox{\\quad for $\\mathbf{X}$ in $R$}.", "$$", "Then", "\\begin{equation}\\label{eq:7.1.35}", "\\Sigma_1(M_j-m_j) V(R_j)\\le2M\\,\\Sigma_1 V(R_j)=2M\\sum_{j=1}^m V(T_j)<", "2M\\epsilon,", "\\end{equation}", "from \\eqref{eq:7.1.32}.", "Since $f$ is uniformly continuous on the compact set $C$", "(Theorem~\\ref{thmtype:5.2.14}),", "there is a $\\delta>0$ such that $M_j-m_j<\\epsilon$ if", "$\\|{\\bf P}\\|< \\delta$ and $R_j\\subset C$; hence,", "$$", "\\Sigma_2(M_j-m_j)V(R_j)<\\epsilon\\Sigma_2\\, V(R_j)\\le\\epsilon V(R).", "$$", "This, \\eqref{eq:7.1.34}, and \\eqref{eq:7.1.35} imply that", "$$", "S({\\bf P})-s({\\bf P})<[2M+V(R)]\\epsilon", "$$", "if $\\|{\\bf P}\\|<\\delta$ and ${\\bf P}$ is a refinement of ${\\bf P}_0$.", "Therefore, Theorem~\\ref{thmtype:7.1.12} implies that $f$ is integrable on", "$R$.", "\\enlargethispage{4\\baselineskip}" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.2.14", "TRENCH_REAL_ANALYSIS-thmtype:7.1.12" ], "ref_ids": [ 154, 195 ] } ], "ref_ids": [] }, { "id": 198, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.19", "categories": [], "title": "", "contents": [ "Suppose that $f$ is bounded on a bounded set $S$ and continuous", "except on a subset $E$ of $S$ with zero content. Suppose also that", "$\\partial S$ has zero content$.$ Then $f$ is integrable on $S.$" ], "refs": [], "proofs": [ { "contents": [ "Let $f_S$ be as in \\eqref{eq:7.1.36}. Since a discontinuity of", "$f_S$ is either a discontinuity of $f$ or a point of $\\partial S$, the", "set of discontinuities of $f_S$ is the union of two sets of zero", "content and therefore is of zero content (Lemma~\\ref{thmtype:7.1.15}).", "Therefore, $f_S$ is integrable on any rectangle containing $S$", "(from Theorem~\\ref{thmtype:7.1.16}), and consequently on $S$", "(Definition~\\ref{thmtype:7.1.17})." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.1.15", "TRENCH_REAL_ANALYSIS-thmtype:7.1.16", "TRENCH_REAL_ANALYSIS-thmtype:7.1.17" ], "ref_ids": [ 261, 197, 362 ] } ], "ref_ids": [] }, { "id": 199, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.21", "categories": [], "title": "", "contents": [ "A differentiable surface in $\\R^n$ has zero content$.$" ], "refs": [], "proofs": [ { "contents": [ "Let $S$, $D$, and $\\mathbf{G}$ be as in Definition~\\ref{thmtype:7.1.20}.", "From Lemma~\\ref{thmtype:6.2.7}, there is a constant $M$ such", "that", "\\begin{equation}\\label{eq:7.1.37}", "|\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})|\\le", "M|\\mathbf{X}-\\mathbf{Y}|\\mbox{\\quad if\\quad}\\mathbf{X},\\mathbf{Y}\\in D.", "\\end{equation}", "Since $D$ is bounded, $D$ is contained in a cube", "$$", "C=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_m,b_m],", "$$", "where", "$$", "b_i-a_i=L,\\quad 1\\le i\\le m.", "$$", "Suppose that we partition $C$ into $N^m$ smaller cubes by partitioning", "each of the intervals $[a_i,b_i]$ into $N$ equal subintervals. Let", "$R_1$, $R_2$, \\dots, $R_k$ be the smaller cubes so produced that", "contain", "points of $D$, and select points $\\mathbf{X}_1$, $\\mathbf{X}_2$, \\dots,", "$\\mathbf{X}_k$", "such that $\\mathbf{X}_i\\in D\\cap R_i$, $1\\le i\\le k$. If $\\mathbf{Y}", "\\in D\\cap R_i$, then \\eqref{eq:7.1.37} implies that", "\\begin{equation}\\label{eq:7.1.38}", "|\\mathbf{G}(\\mathbf{X}_i)-\\mathbf{G}(\\mathbf{Y})|\\le M|\\mathbf{X}_i-\\mathbf{Y}|.", "\\end{equation}", "Since $\\mathbf{X}_i$ and $\\mathbf{Y}$ are both in the cube $R_i$ with", "edge length $L/N$,", "$$", "|\\mathbf{X}_i-\\mathbf{Y}|\\le\\frac{L\\sqrt{m}}{ N}.", "$$", " This and \\eqref{eq:7.1.38} imply that", "$$", "|\\mathbf{G}(\\mathbf{X}_i)-\\mathbf{G}(\\mathbf{Y})|\\le\\frac{ML\\sqrt m}{ N},", "$$", "which in turn implies that", "$\\mathbf{G}(\\mathbf{Y})$ lies in a cube $\\widetilde{R}_i$ in $\\R^n$", " centered at $\\mathbf{G}(\\mathbf{X}_i)$,", "with", "sides of length $2ML\\sqrt{m}/N$.", " Now", "$$", "\\sum_{i=1}^k V(\\widetilde{R}_i)= k\\left(\\frac{2ML\\sqrt{m}}{", "N}\\right)^n\\le", "N^m\\left(\\frac{2ML\\sqrt{m}}{ N}\\right)^n=(2ML\\sqrt{m})^n", "N^{m-n}.", "$$", "Since $n>m$, we can make the sum on the left arbitrarily small by", "taking $N$ sufficiently large. Therefore, $S$ has zero content." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.1.20", "TRENCH_REAL_ANALYSIS-thmtype:6.2.7" ], "ref_ids": [ 364, 258 ] } ], "ref_ids": [] }, { "id": 200, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.22", "categories": [], "title": "", "contents": [ "Suppose that $S$ is a bounded set in $\\R^n,$ with boundary", "consisting of a finite number of differentiable surfaces$.$ Let $f$ be", "bounded on $S$ and continuous except on a set of zero content. Then", "$f$ is integrable on $S.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 201, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.23", "categories": [], "title": "", "contents": [ "If $f$ and $g$ are integrable on $S,$ then so is $f+g,$ and", "$$", "\\int_S(f+g)(\\mathbf{X})\\,d\\mathbf{X}=\\int_S f(\\mathbf{X})\\,d\\mathbf{X}+", "\\int_S g(\\mathbf{X})\\,d\\mathbf{X}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.20}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 202, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.24", "categories": [], "title": "", "contents": [ "If $f$ is integrable on $S$ and $c$ is a constant$,$ then $cf$ is", "integrable on $S,$ and", "$$", "\\int_S(cf)(\\mathbf{X})\\,d\\mathbf{X}=c\\int_S f(\\mathbf{X})\\,d\\mathbf{X}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.21}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 203, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.25", "categories": [], "title": "", "contents": [ "If $f$ and $g$ are integrable on $S$ and $f(\\mathbf{X})\\le g(\\mathbf{X})$", "for $\\mathbf{X}$ in $S,$ then", "$$", "\\int_S f(\\mathbf{X})\\,d\\mathbf{X}\\le\\int_S g(\\mathbf{X})\\,d\\mathbf{X}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.22}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 204, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.26", "categories": [], "title": "", "contents": [ " If $f$ is integrable on $S,$", "then so is $|f|,$ and", "$$", "\\left|\\int_S f(\\mathbf{X})\\,d\\mathbf{X}\\right|\\le\\int_S |f(\\mathbf{X})|\\,d\\mathbf{X}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.23}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 205, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.27", "categories": [], "title": "", "contents": [ "If $f$ and $g$ are integrable on $S,$ then so is the product $fg.$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.24}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 206, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.28", "categories": [], "title": "", "contents": [ "Suppose that $u$ is continuous and $v$ is integrable and nonnegative", "on a rectangle $R.$ Then", "$$", "\\int_R u(\\mathbf{X})v(\\mathbf{X})\\,d\\mathbf{X}=", "u(\\mathbf{X}_0)\\int_R v(\\mathbf{X})\\,d\\mathbf{X}", "$$", "for some $\\mathbf{X}_0$ in $R.$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.25}.", "\\begin{lemma}\\label{thmtype:7.1.29}", "Suppose that $S$ is contained in a bounded set $T$ and $f$ is integrable", "on $S.$ Then", " $f_S$ $($see $\\eqref{eq:7.1.36})$ is integrable on $T,$ and", "$$", "\\int_T f_S(\\mathbf{X})\\,d\\mathbf{X}=\\int_S f(\\mathbf{X})\\,d\\mathbf{X}.", "$$", "\\end{lemma}", "\\nopagebreak" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 207, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.30", "categories": [], "title": "", "contents": [ "If $f$ is integrable on disjoint sets $S_1$ and $S_2,$ then $f$ is", "integrable on $S_1\\cup S_2,$ and", "\\begin{equation}\\label{eq:7.1.39}", "\\int_{S_1\\cup S_2} f(\\mathbf{X})\\,d\\mathbf{X}=", "\\int_{S_1} f(\\mathbf{X})\\,d\\mathbf{X}+", "\\int_{S_2} f(\\mathbf{X})\\,d\\mathbf{X}.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "For $i=1$, $2$, let", "$$", "f_{S_i}(\\mathbf{X})=\\left\\{\\casespace\\begin{array}{ll} f(\\mathbf{X}),&\\mathbf{X}\\in", "S_i,\\\\[2\\jot]", " 0,&\\mathbf{X}\\not\\in S_i.\\end{array}\\right.", "$$", "From Lemma~\\ref{thmtype:7.1.29} with $S=S_i$ and $T=S_1\\cup S_2$,", "$f_{S_i}$ is integrable on $S_1\\cup S_2$, and", "$$", "\\int_{S_1\\cup S_2} f_{S_i}(\\mathbf{X})\\,d\\mathbf{X}", "=\\int_{S_i} f(\\mathbf{X})\\,d\\mathbf{X},\\quad i=1,2.", "$$", "Theorem~\\ref{thmtype:7.1.23} now implies that $f_{S_1}+f_{S_2}$ is integrable on", "$S_1\\cup S_2$ and", "\\begin{equation}\\label{eq:7.1.40}", "\\int_{S_1\\cup S_2} (f_{S_1}+f_{S_2})(\\mathbf{X})\\,d\\mathbf{X}=\\int_{S_1}", "f(\\mathbf{X})\\,d\\mathbf{X}+\\int_{S_2} f(\\mathbf{X})\\, d\\mathbf{X}.", "\\end{equation}", "\\newpage", "\\noindent", "Since $S_1\\cap S_2=\\emptyset$,", "$$", "\\left(f_{S_1}+f_{S_2}\\right)(\\mathbf{X})=", "f_{S_1}(\\mathbf{X})+f_{S_2}(\\mathbf{X})", "=f(\\mathbf{X}),\\quad \\mathbf{X}\\in S_1\\cup S_2.", "$$", " Therefore,", "\\eqref{eq:7.1.40} implies \\eqref{eq:7.1.39}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.1.29", "TRENCH_REAL_ANALYSIS-thmtype:7.1.23" ], "ref_ids": [ 262, 201 ] } ], "ref_ids": [] }, { "id": 208, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.2.1", "categories": [], "title": "", "contents": [ "$R= [a,b]\\times [c,d]$ and", "$$", " F(y)=\\int_a^b f(x,y)\\,dx", "$$", "exists for each $y$ in $[c,d].$ Then $F$ is integrable on $[c,d],$", "and", "\\begin{equation}\\label{eq:7.2.1}", "\\int_c^d F(y)\\,dy=\\int_R f(x,y)\\,d(x,y);", "\\end{equation}", "that is$,$", "\\begin{equation}\\label{eq:7.2.2}", "\\int_c^d dy\\int_a^b f(x,y)\\,dx=\\int_R f(x,y)\\,d(x,y).", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Let", "$$", "P_1: a=x_00$ a", "partition $\\mathbf{P}$ of $R$ such that $S_f(\\mathbf{P})-s_f(\\mathbf{P})<\\epsilon$,", "from Theorem~\\ref{thmtype:7.1.12}. Consequently, from", "\\eqref{eq:7.2.6}, there is", "a partition $P_2$ of $[c,d]$ such that", "$S_F(P_2)-s_F(P_2)<\\epsilon$,", " so $F$ is integrable on $[c,d]$, from", "Theorem~\\ref{thmtype:3.2.7}.", "It remains to verify \\eqref{eq:7.2.1}. From \\eqref{eq:7.2.4} and the", "definition of $\\int_c^dF(y)\\,dy$,", "there is for each $\\epsilon>0$ a $\\delta>0$ such that", "$$", "\\left|\\int_c^d F(y)\\,dy-\\sigma\\right|<\\epsilon\\mbox{\\quad if\\quad}", "\\|P_2\\|<\\delta;", "$$", "that is,", "$$", "\\sigma-\\epsilon<\\int_c^d F(y)\\,dy<\\sigma+\\epsilon\\mbox{\\quad if \\quad}", "\\|P_2\\|<\\delta.", "$$", "This and \\eqref{eq:7.2.5} imply that", "$$", "s_f(\\mathbf{P})-\\epsilon<\\int_c^d F(y)\\,dy0$ a", "partition $\\mathbf{P}$ of $R$ such that $S_f(\\mathbf{P})-s_f(\\mathbf{P})<\\epsilon$,", "from Theorem~\\ref{thmtype:7.1.12}. Consequently, from \\eqref{eq:7.2.11},", "there", "is a partition $\\mathbf{Q}$ of $T$ such that", "$S_{F_p}(\\mathbf{Q})-s_{F_p}(\\mathbf{Q})<\\epsilon$, so $F_p$ is integrable", "on $T$, from Theorem~\\ref{thmtype:7.1.12}.", "It remains to verify that", "\\begin{equation} \\label{eq:7.2.12}", "\\int_R f(\\mathbf{X})\\,d\\mathbf{X}=", "\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}.", "\\end{equation}", "From \\eqref{eq:7.2.9} and the definition of $\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}$, there", "is for each $\\epsilon>0$ a $\\delta>0$ such that", "$$", "\\left|\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", "-\\sigma\\right|<\\epsilon\\mbox{\\quad", "if\\quad}", "\\|\\mathbf{Q}\\|<\\delta;", "$$", "that is,", "$$", "\\sigma-\\epsilon<\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", "<\\sigma+", "\\epsilon\\mbox{\\quad if \\quad}\\|\\mathbf{Q}\\|<\\delta.", "$$", "This and \\eqref{eq:7.2.10} imply that", "$$", "s_f(\\mathbf{P})-\\epsilon<", "\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", "2$ and the proposition is true with $n$ replaced", "by $n-1$. Holding $x_n$ fixed and applying this assumption", "yields", "$$", "F_n(x_n)=", "\\int^{b_{n-1}}_{a_{n-1}}", "dx_{n-1}\\int_{a_{n-2}}^{b_{n-2}}dx_{n-2}\\cdots", "\\int^{b_2}_{a_2} dx_2\\int^{b_1}_{a_1} f(\\mathbf{X})\\,dx_1.", "$$", "Now Theorem~\\ref{thmtype:7.2.3} with $p=n-1$ completes the induction." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.2.1", "TRENCH_REAL_ANALYSIS-thmtype:7.2.3" ], "ref_ids": [ 208, 209 ] } ], "ref_ids": [] }, { "id": 211, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.2.5", "categories": [], "title": "", "contents": [ "If $f$ is continuous on", "$$", "R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n],", "$$", "then $\\int_R f(\\mathbf{X})\\,d\\mathbf{X}$ can be evaluated by iterated", "integrals in any of the $n!$ ways indicated in $\\eqref{eq:7.2.16}.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 212, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.2.6", "categories": [], "title": "", "contents": [ "If $f$ is integrable on the set $S$ in $\\eqref{eq:7.2.17}$ and the", "integral $\\eqref{eq:7.2.19}$ exists for $c\\le y\\le d,$ then", "\\begin{equation}\\label{eq:7.2.20}", "\\int_S f(x,y) \\,d(x,y)=\\int_c^d dy\\int^{v(y)}_{u(y)} f(x,y)\\,dx.", "\\end{equation}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 213, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.2.7", "categories": [], "title": "", "contents": [ "Suppose that $f$ is integrable on", "$$", "S=\\set{(x,y,z)}{u_1(y,z)\\le x\\le v_1(y,z),\\ u_2(z)\\le y\\le v_2(z),\\", "c\\le z\\le d},", "$$", "and let", "$$", "S(z)=\\set{(x,y)}{u_1(y,z)\\le x\\le v_1(y,z),\\ u_2(z)\\le y\\le v_2(z)}", "$$", "for each $z$ in $[c,d].$ Then", "$$", "\\int_S f(x,y,z)\\,d(x,y,z)=\\int_c^d dz\\int^{v_2(z)}_{u_2(z)} dy", "\\int^{v_1(y,z)}_{u_1(y,z)} f(x,y,z)\\,dx,", "$$", "provided that", "$$", "\\int^{v_1(y,z)}_{u_1(y,z)} f(x,y,z)\\,dx", "$$", "exists for all $(y,z)$ such that", "$$", "c\\le z\\le d\\mbox{\\quad and\\quad} u_2(z)\\le y\\le v_2(z),", "$$", "and", "$$", "\\int_{S(z)} f(x,y,z)\\,d(x,y)", "$$", "exists for all $z$ in $[c,d].$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 214, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.1", "categories": [], "title": "", "contents": [ "A bounded set $S$ is Jordan measurable if and only if the boundary", "of $S$ has", "zero content$.$" ], "refs": [], "proofs": [ { "contents": [ "Let $R$ be a rectangle containing $S$. Suppose that $V(\\partial S)=0$.", "Since", "$\\psi_{S}$ is bounded on $R$ and discontinuous only on", "$\\partial S$", "(Exercise~\\ref{exer:2.2.9}), Theorem~\\ref{thmtype:7.1.19}", "implies that $\\int_R\\psi_S (\\mathbf{X})\\,d\\mathbf{X}$ exists.", " For the converse, suppose that", "$\\partial S$ does not have zero content", "and let ${\\bf P}=\\{R_1, R_2,\\dots, R_k\\}$ be a partition", "of $R$. For each $j$ in $\\{1,2,\\dots,k\\}$ there are three", "possibilities:", "\\begin{description}", " \\item{1.} $R_j\\subset S$; then", "$$", "\\min\\set{\\psi_S(\\mathbf{X})}{\\mathbf{X}\\in R_j}=", "\\max\\set{\\psi_S(\\mathbf{X})}{\\mathbf{X}\\in R_j}=1.", "$$", "\\item{2.} $R_j\\cap S\\ne\\emptyset$ and $R_j\\cap S^c\\ne", "\\emptyset$; then", "$$", "\\min\\set{\\psi_S (\\mathbf{X})}{\\mathbf{X}\\in R_j}=0\\mbox{\\quad and\\quad}", "\\max\\set{\\psi_S(\\mathbf{X})}{\\mathbf{X}\\in R_j}=1.", "$$", "\\item{3.} $R_j\\subset S^c$; then", "$$", "\\min\\set{\\psi_S(\\mathbf{X})}{\\mathbf{X}\\in R_j}=\\max\\set{\\psi_S(\\mathbf{X})}", "{\\mathbf{X}\\in R_j}=0.", "$$", "\\end{description}", "\\newpage", "\\noindent Let", "\\begin{equation} \\label{eq:7.3.2}", "{\\mathcal U}_1=\\set{j}{R_j\\subset S}", "\\mbox{\\quad and \\quad}", "{\\mathcal U}_2=\\set{j}{R_j\\cap S\\ne\\emptyset\\mbox{ and }R_j\\cap", "S^c\\ne\\emptyset}.", "\\end{equation}", "Then the upper and lower", "sums of $\\psi_S$ over ${\\bf P}$ are", "\\begin{equation}\\label{eq:7.3.3}", "\\begin{array}{rcl}", "S({\\bf P})\\ar=\\dst\\sum_{j\\in{\\mathcal U}_1} V(R_j)+\\sum_{j\\in{\\mathcal U}_2}", "V(R_j)\\\\[2\\jot]", "\\ar=\\mbox{total content of the subrectangles in ${\\bf P}$ that intersect", "$S$}", "\\end{array}", "\\end{equation}", "and", "\\begin{equation}\\label{eq:7.3.4}", "\\begin{array}{rcl}", "s({\\bf P})\\ar=\\dst\\sum_{j\\in{\\mathcal U}_1} V(R_j) \\\\", "\\ar=\\mbox{total content of the subrectangles in ${\\bf P}$", "contained in $S$}.", "\\end{array}", "\\end{equation}", "Therefore,", "$$", "S({\\bf P})-s({\\bf P})=\\sum_{j\\in {\\mathcal U}_2} V(R_j),", "$$", "which is the total content of the subrectangles in ${\\bf P}$ that", "intersect both $S$ and $S^c$.", " Since these subrectangles contain", "$\\partial S$,", "which does not have zero content, there is an", "$\\epsilon_0>0$ such that", "$$", "S({\\bf P})-s({\\bf P})\\ge\\epsilon_0", "$$", "for every partition ${\\bf P}$ of $R$. By", "Theorem~\\ref{thmtype:7.1.12}, this implies that $\\psi_S$ is not", "integrable on $R$, so $S$ is not Jordan measurable." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.1.19", "TRENCH_REAL_ANALYSIS-thmtype:7.1.12" ], "ref_ids": [ 198, 195 ] } ], "ref_ids": [] }, { "id": 215, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.5", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{G}:\\R^n\\to \\R^n$ is regular on a compact", "Jordan measurable set $S.$ Then $\\mathbf{G}(S)$ is compact and", "Jordan measurable$.$" ], "refs": [], "proofs": [ { "contents": [ "We leave it to you to prove that $\\mathbf{G}(S)$ is", "compact", "(Exercise~6.2.23). Since $S$ is", "Jordan measurable,", " $V(\\partial S)=0$, by Theorem~\\ref{thmtype:7.3.1}.", "Therefore, $V(\\mathbf{G}(\\partial S))=0$, by Lemma~\\ref{thmtype:7.3.4}.", "But $\\mathbf{G}(\\partial S)=", "\\partial(\\mathbf{G}(S))$ (Exercise~\\ref{exer:6.3.23}), so", "$V(\\partial(\\mathbf{G}(S)))=0$, which implies that", "$\\mathbf{G}(S)$ is Jordan measurable, again by Theorem~\\ref{thmtype:7.3.1}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.3.1", "TRENCH_REAL_ANALYSIS-thmtype:7.3.4", "TRENCH_REAL_ANALYSIS-thmtype:7.3.1" ], "ref_ids": [ 214, 264, 214 ] } ], "ref_ids": [] }, { "id": 216, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.7", "categories": [], "title": "", "contents": [ "If $S$ is a compact Jordan measurable subset", " of $\\R^n$ and $\\mathbf{L}:\\R^n\\to \\R^n$ is the invertible linear", "transformation", "$\\mathbf{X}=\\mathbf{L}(\\mathbf{Y})=\\mathbf{AY},$ then", "\\begin{equation}\\label{eq:7.3.14}", "V(\\mathbf{L}(S))=|\\det(\\mathbf{A})| V(S).", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Theorem~\\ref{thmtype:7.3.5} implies that $\\mathbf{L}(S)$ is", "Jordan measurable. If", "\\begin{equation} \\label{eq:7.3.15}", "V(\\mathbf{L}(R))=|\\det(\\mathbf{A})| V(R)", "\\end{equation}", "whenever $R$ is a rectangle, then", " \\eqref{eq:7.3.14} holds if $S$", "is any compact Jordan measurable set. To see this, suppose that", "$\\epsilon>0$, let", "$R$ be a rectangle containing $S$, and let", "${\\bf P}=\\{R_1,R_2,\\dots,R_k\\}$ be a partition of $R$ such that the", "upper and lower sums of $\\psi_S$ over ${\\bf", "P}$ satisfy the inequality", "\\begin{equation}\\label{eq:7.3.16}", "S({\\bf P})-s({\\bf P})<\\epsilon.", "\\end{equation}", "Let ${\\mathcal U}_1$ and ${\\mathcal U}_2$ be as in \\eqref{eq:7.3.2}.", "From \\eqref{eq:7.3.3} and \\eqref{eq:7.3.4},", "\\begin{equation}\\label{eq:7.3.17}", "s({\\bf P})=\\sum_{j\\in{\\mathcal U}_1} V(R_j)\\le V(S)\\le\\sum_{j\\in{\\mathcal U}_1} V(R_j)+\\sum_{j\\in{\\mathcal U}_2}", "V(R_j)=S({\\bf P}).", "\\end{equation}", " Theorem~\\ref{thmtype:7.3.7}", "implies that $\\mathbf{L}(R_1)$, $\\mathbf{L}(R_2)$, \\dots, $\\mathbf{L}(R_k)$", "and", "$\\mathbf{L}(S)$ are all Jordan measurable.", "Since", "$$", "\\bigcup_{j\\in{\\mathcal U}_1}R_j\\subset S\\subset\\bigcup_{j\\in{\\mathcal", "S}_1\\cup{\\mathcal S_2}}R_j,", "$$", "it follows that", "$$", "L\\left(\\bigcup_{j\\in{\\mathcal U}_1}R_j\\right)\\subset", "L(S)\\subset L\\left(\\bigcup_{j\\in{\\mathcal S}_1\\cup{\\mathcal S_2}}R_j\\right).", "$$", "Since $L$ is one-to-one on $\\R^n$, this implies that", "\\begin{equation} \\label{eq:7.3.18}", "\\sum_{j\\in{\\mathcal U}_1} V(\\mathbf{L}(R_j))\\le V(\\mathbf{L}(S))\\le\\sum_{j\\in{\\mathcal U}_1}", "V(\\mathbf{L}(R_j))+\\sum_{j\\in{\\mathcal U}_2} V(\\mathbf{L}(R_j)).", "\\end{equation}", "If we assume that \\eqref{eq:7.3.15} holds whenever $R$ is a rectangle,", "then", "$$", "V(\\mathbf{L}(R_j))=|\\det(\\mathbf{A})|V(R_j),\\quad 1\\le j\\le k,", "$$", "so \\eqref{eq:7.3.18} implies that", "$$", "s({\\bf P})\\le \\frac{V(\\mathbf{L}(S))}{ |\\det(\\mathbf{A})|}\\le S({\\bf P}).", "$$", "This, \\eqref{eq:7.3.16} and \\eqref{eq:7.3.17} imply that", "$$", "\\left|V(S)-\\frac{V(\\mathbf{L}(S))}{ |\\det(\\mathbf{A})|}\\right|<\\epsilon;", "$$", "hence, since $\\epsilon$ can be made arbitrarily small, \\eqref{eq:7.3.14}", "follows for any Jordan measurable set.", "To complete the proof, we must verify \\eqref{eq:7.3.15} for every", "rectangle", "$$", "R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n]=I_1\\times", "I_2\\times\\cdots\\times I_n.", "$$", " Suppose that $\\mathbf{A}$ in \\eqref{eq:7.3.12} is an elementary matrix;", "that is, let", "$$", "\\mathbf{X}=\\mathbf{L}(\\mathbf{Y})=\\mathbf{EY}.", "$$", "{\\sc Case 1}. If $\\mathbf{E}$ is obtained by interchanging the $i$th and", "$j$th rows of $\\mathbf{I}$, then", "$$", "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$ and $r\\ne j$};\\\\", "y_j&\\mbox{if $r=i$};\\\\", "y_i&\\mbox{if $r=j$}.\\end{array}\\right.", "$$", "Then $\\mathbf{L}(R)$ is the Cartesian product of $I_1$,", "$I_2$, \\dots, $I_n$ with", "$I_i$ and $I_j$ interchanged, so", "$$", "V(\\mathbf{L}(R))=V(R)=|\\det(\\mathbf{E})|V(R)", "$$", "since $\\det(\\mathbf{E})=-1$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", "{\\sc Case 2}. If $\\mathbf{E}$ is obtained by multiplying the $r$th row of", "$\\mathbf{I}$ by $a$, then", "$$", "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$},\\\\", "ay_i&\\mbox{if $r=i$}.\\end{array}\\right.", "$$", "Then", "$$", "\\mathbf{L}(R)=I_1\\times\\cdots\\times I_{i-1}\\times I'_i\\times I_{i+1}\\times", "\\cdots\\times I_n,", "$$", "where $I'_i$ is an interval with length equal to $|a|$ times the", "length of $I_i$, so", "$$", "V(\\mathbf{L}(R))=|a|V(R)=|\\det(\\mathbf{E})|V(R)", "$$", "since $\\det(\\mathbf{E})=a$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", "{\\sc Case 3}. If $\\mathbf{E}$ is obtained by adding $a$ times the $j$th", "row of $\\mathbf{I}$ to its $i$th row ($j\\ne i$), then", "$$", "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$};\\\\", "y_i+ay_j&\\mbox{if $r=i$}.\\end{array}\\right.", "$$", "Then", "$$", "\\mathbf{L}(R)=\\set{(x_1,x_2,\\dots,x_n)}{a_i+ax_j\\le x_i\\le b_i+ax_j", "\\mbox{ and } a_r\\le x_r\\le b_r\\mbox{if } r\\ne i},", "$$", "which is a parallelogram if $n=2$ and a parallelepiped if $n=3$", "(Figure~\\ref{figure:7.3.1}). Now", "$$", "V(\\mathbf{L}(R))=\\int_{\\mathbf{L}(R)} d\\mathbf{X},", "$$", "which we can evaluate as an iterated integral in which the first", "integration is with respect to $x_i$. For example, if $i=1$, then", "\\begin{equation}\\label{eq:7.3.19}", "V(\\mathbf{L}(R))=\\int^{b_n}_{a_n} dx_n\\int^{b_{n-1}}_{a_{n-1}}", "dx_{n-1}\\cdots\\int^{b_2}_{a_2} dx_2\\int^{b_1+ax_j}_{a_1+ax_j} dx_1.", "\\end{equation}", "\\newpage", "\\noindent", "Since", "$$", "\\int^{b_1+ax_j}_{a_1+ax_j} dy_1=\\int^{b_1}_{a_1} dy_1,", "$$", "\\eqref{eq:7.3.19} can be rewritten as", "\\begin{eqnarray*}", "V(\\mathbf{L}(R))\\ar=\\int^{b_n}_{a_n} dx_n\\int^{b_{n-1}}_{a_{n-1}}", "dx_{n-1}\\cdots\\int^{b_2}_{a_2} dx_2\\int^{b_1}_{a_1} dx_1\\\\", "\\ar=(b_n-a_n)(b_{n-1}-a_{n-1})\\cdots (b_1-a_1)=V(R).", "\\end{eqnarray*}", " Hence,", "$V(\\mathbf{L}(R))=|\\det(\\mathbf{E})|V(R)$,", "since $\\det(\\mathbf{E})=1$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", "\\vskip12pt", " \\centereps{3.6in}{4.6in}{fig070301.eps}", " \\vskip6pt", " \\refstepcounter{figure}", " \\centerline{\\bf Figure \\thefigure} \\label{figure:7.3.1}", " \\vskip12pt", "From what we have shown so far, \\eqref{eq:7.3.14} holds if $\\mathbf{A}$ is an", "elementary matrix and $S$ is any compact Jordan measurable set. If", "$\\mathbf{A}$ is an arbitrary nonsingular matrix,", "\\newpage", "\\noindent", "\\hskip -.0em", "then we can write $\\mathbf{A}$", "as a product of elementary matrices \\eqref{eq:7.3.10} and apply our known", "result successively to $\\mathbf{L}_1$, $\\mathbf{L}_2$, \\dots, $\\mathbf{L}_k$", "(see", "\\eqref{eq:7.3.13}). This yields", "$$", "V(\\mathbf{L}(S))=|\\det(\\mathbf{E}_k)|\\,|\\det(\\mathbf{E}_{k-1})|\\cdots", "|\\det\\mathbf{E}_1| V(S)=|\\det(\\mathbf{A})|V(S),", "$$", "by Theorem~\\ref{thmtype:6.1.9} and induction." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.3.5", "TRENCH_REAL_ANALYSIS-thmtype:7.3.7", "TRENCH_REAL_ANALYSIS-thmtype:6.1.9" ], "ref_ids": [ 215, 216, 173 ] } ], "ref_ids": [] }, { "id": 217, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.8", "categories": [], "title": "", "contents": [ "\\E^n\\to \\R^n$ is regular on a compact Jordan measurable set $S$ and", "$f$ is continuous on $\\mathbf{G}(S).$ Then", "\\begin{equation}\\label{eq:7.3.28}", "\\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}=", "\\int_S f(\\mathbf{G}(\\mathbf{Y}))", "|J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Let $s$ be the edge length of $C$. Let $\\mathbf{Y}_0=", "(c_1,c_2,\\dots,c_n)$ be the center of $C$, and suppose that", " $\\mathbf{H}=(y_1,y_2,\\dots,y_n)\\in C$.", "If $\\mathbf{H}= (h_1,h_2,\\dots,h_n)$ is continuously differentiable on", "$C$, then applying the mean value theorem", "(Theorem~\\ref{thmtype:5.4.5}) to the components of", "$\\mathbf{H}$ yields", "$$", "h_i(\\mathbf{Y})-h_i(\\mathbf{Y}_0)=\\sum_{j=1}^n", "\\frac{\\partial h_i(\\mathbf{Y}_i)}{\\partial y_j}(y_j-c_j),\\quad 1\\le i\\le n,", "$$", "where $\\mathbf{Y}_i\\in C$. Hence, recalling that", "$$", "\\mathbf{H}'(\\mathbf{Y})=\\left[\\frac{\\partial h_i}{\\partial", "y_j}\\right]_{i,j=1}^n,", "$$", "applying Definition~\\ref{thmtype:7.3.9}, and noting that $|y_j-c_j|\\le", "s/2$, $1\\le j\\le n$, we infer that", "$$", "|h_i(\\mathbf{Y})-h_i(\\mathbf{Y}_0)|\\le \\frac{s}{2}", "\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C},\\quad 1\\le i\\le", "n.", "$$", "This means that $\\mathbf{H}(C)$ is", "contained in a cube with center $\\mathbf{X}_0=\\mathbf{H}(\\mathbf{Y}_0)$ and edge", " length", "$$", "s\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C}.", "$$", "Therefore,", "\\begin{equation}\\label{eq:7.3.30}", "\\begin{array}{rcl}", "V(\\mathbf{H}(C))\\ar\\le", "\\left[\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty\\right]^n}{\\mathbf{Y}\\in", "C} s^n\\\\[2\\jot]", "\\ar=\\left[\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty\\right]^n}{\\mathbf{Y}\\in C}", "V(C).", "\\end{array}", "\\end{equation}", "Now let", "$$", "\\mathbf{L}(\\mathbf{X})=\\mathbf{A}^{-1}\\mathbf{X}", "$$", "and set $\\mathbf{H}=\\mathbf{L}\\circ\\mathbf{G}$; then", "$$", "\\mathbf{H}(C)=\\mathbf{L}(\\mathbf{G}(C))", "\\mbox{\\quad and\\quad}\\mathbf{H}'=\\mathbf{A}^{-1}\\mathbf{G}',", "$$", "so \\eqref{eq:7.3.30} implies that", "\\begin{equation}\\label{eq:7.3.31}", "V(\\mathbf{L}(\\mathbf{G}(C)))\\le", "\\left[\\max\\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C}", "\\right]^nV(C).", "\\end{equation}", "Since $\\mathbf{L}$ is linear,", "Theorem~\\ref{thmtype:7.3.7} with $\\mathbf{A}$ replaced by $\\mathbf{A}^{-1}$ implies that", "$$", "V(\\mathbf{L}(\\mathbf{G}(C)))=|\\det(\\mathbf{A})^{-1}|V(\\mathbf{G}(C)).", "$$", "This and \\eqref{eq:7.3.31} imply that", "$$", "|\\det(\\mathbf{A}^{-1})|V(\\mathbf{G}(C))", "\\le\\left[\\max\\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in", "C}", "\\right]^nV(C).", "$$", "Since $\\det(\\mathbf{A}^{-1})=1/\\det(\\mathbf{A})$, this", "implies \\eqref{eq:7.3.29}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.4.5", "TRENCH_REAL_ANALYSIS-thmtype:7.3.9", "TRENCH_REAL_ANALYSIS-thmtype:7.3.7" ], "ref_ids": [ 164, 365, 216 ] } ], "ref_ids": [] }, { "id": 218, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.15", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{G}: \\E^n\\to \\R^n$ is continuously", "differentiable on a bounded open set $N$ containing the compact", "Jordan measurable set $S,$ and regular on $S^0.$ Suppose also that", "$\\mathbf{G}(S)$ is Jordan measurable$,$", "$f$ is continuous on $\\mathbf{G}(S),$ and $G(C)$ is Jordan measurable for", "every cube $C\\subset N$. Then", "\\begin{equation}\\label{eq:7.3.50}", "\\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}=", "\\int_S f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Since $f$ is continuous on $\\mathbf{G}(S)$ and", " $(|J\\mathbf{G}|) f\\circ\\mathbf{G}$ is continuous on $S$, the integrals", "in \\eqref{eq:7.3.50} both exist, by", "Corollary~\\ref{thmtype:7.3.2}.", "Now let", "$$", "\\rho=\\dist\\ (\\partial S, N^c)", "$$", "(Exercise~5.1.25), and", "$$", "P=\\set{\\mathbf{Y}}{\\dist(\\mathbf{Y}, \\partial S)}\\le", "\\frac{\\rho}{2}.", "$$", " Then $P$ is a", "compact subset of $N$ (Exercise~5.1.26) and", "$\\partial S\\subset P^0$", "(Figure~\\ref{figure:7.3.4}).", " Since $S$ is Jordan measurable, $V(\\partial S)=0$, by", "Theorem~\\ref{thmtype:7.3.1}. Therefore,", "if $\\epsilon>0$, we can choose cubes $C_1$, $C_2$, \\dots, $C_k$", " in $P^0$ such that", "\\begin{equation} \\label{eq:7.3.51}", "\\partial S\\subset\\bigcup_{j=1}^k C_j^0", "\\end{equation}", "and", "\\begin{equation}\\label{eq:7.3.52}", "\\sum_{j=1}^k V(C_j)<\\epsilon", "\\end{equation}", " Now let $S_1$ be the closure of the set of points in $S$", "that are not in any of the cubes $C_1$, $C_2$, \\dots, $C_k$; thus,", "$$", "S_1=\\overline{S\\cap\\left(\\cup_{j=1}^k C_j\\right)^c}.", "$$", "\\newpage", "\\noindent", "Because of \\eqref{eq:7.3.51}, $S_1\\cap \\partial S=\\emptyset$,", "so $S_1$ is a compact Jordan measurable subset of $S^0$. Therefore,", "$\\mathbf{G}$ is regular on $S_1$, and $f$ is continuous on", "$\\mathbf{G}(S_1)$.", "Consequently, if $Q$ is as defined in \\eqref{eq:7.3.37}, then $Q(S_1)=0$", "by Theorem~\\ref{thmtype:7.3.8}.", " \\vskip12pt", " \\centereps{2.1in}{2.8in}{fig070304.eps}", " \\vskip6pt", " \\refstepcounter{figure}", " \\centerline{\\bf Figure \\thefigure} \\label{figure:7.3.4}", " \\vskip12pt", "Now", "\\begin{equation}\\label{eq:7.3.53}", "Q(S)=Q(S_1)+Q(S\\cap S_1^c)=Q(S\\cap S_1^c)", "\\end{equation}", "(Exercise~\\ref{exer:7.3.11}) and", "$$", "|Q(S\\cap S_1^c)|\\le\\left|\\int_{\\mathbf{G}(S\\cap S_1^c)} f(\\mathbf{X})\\,d\\mathbf{X}\\right|+\\left|", "\\int_{S\\cap S_1^c} f(\\mathbf{G}(\\mathbf{Y}))|J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\right|.", "$$", " But", "\\begin{equation} \\label{eq:7.3.54}", "\\left|\\int_{S\\cap S_1^c} f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,", "d\\mathbf{Y}\\right|\\le M_1M_2 V(S\\cap S_1^c),", "\\end{equation}", "where $M_1$ and $M_2$ are as defined in \\eqref{eq:7.3.38} and", "\\eqref{eq:7.3.39}. Since", "$S\\cap S_1^c\\subset \\cup_{j=1}^k C_j$,", "\\eqref{eq:7.3.52} implies that $V(S\\cap S_1^k)<\\epsilon$; therefore,", "\\begin{equation} \\label{eq:7.3.55}", "\\left|\\int_{S\\cap S_1^c} f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,", "d\\mathbf{Y}\\right|\\le M_1M_2\\epsilon,", "\\end{equation}", "from \\eqref{eq:7.3.54}. Also", "\\begin{equation}\\label{eq:7.3.56}", "\\left|\\int_{\\mathbf{G}(S\\cap S_1^c)} f(\\mathbf{X})\\,d\\mathbf{X}\\right|\\le M_2", "V(\\mathbf{G}(S\\cap S_1^c))\\le M_2\\sum_{j=1}^k V(\\mathbf{G}(C_j)).", "\\end{equation}", "\\newpage", "\\noindent", "By the argument that led to \\eqref{eq:7.3.30} with", "${\\bf H}={\\bf G}$ and $C=C_{j}$,", "$$", "V(\\mathbf{G}(C_j))\\le\\left[\\max\\set{\\|\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}", "{\\mathbf{Y}\\in C_j}\\right]^nV(C_j),", "$$", "so \\eqref{eq:7.3.56} can be rewritten as", "$$", "\\left|\\int_{\\mathbf{G}(S\\cap S_1^c)} f(\\mathbf{X})\\,d\\mathbf{X}\\right|\\le M_2", "\\left[\\max\\set{\\|\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in P}", "\\right]^n\\epsilon,", "$$", "because of \\eqref{eq:7.3.52}. Since $\\epsilon$ can be made arbitrarily", "small, this and \\eqref{eq:7.3.55} imply that $Q(S\\cap S_1^c)=0$. Now", "$Q(S)=0$, from \\eqref{eq:7.3.53}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.3.2", "TRENCH_REAL_ANALYSIS-thmtype:7.3.1", "TRENCH_REAL_ANALYSIS-thmtype:7.3.8" ], "ref_ids": [ 297, 214, 217 ] } ], "ref_ids": [] }, { "id": 219, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.4", "categories": [], "title": "", "contents": [ "If $(A,N)$ is a normed vector space$,$ then", "\\begin{equation} \\label{eq:8.1.1}", "\\rho(x,y)=N(x-y)", "\\end{equation}", "is a metric on $A.$" ], "refs": [], "proofs": [ { "contents": [ "From \\part{a} with $u=x-y$, $\\rho(x,y)=N(x-y)\\ge0$, with equality", "if and only if $x=y$. From \\part{b} with $u=x-y$ and $a=-1$,", "$$", "\\rho(y,x)=N(y-x)=N(-(x-y))=N(x-y)=\\rho(x,y).", "$$", "From \\part{c} with $u=x-z$ and $v=z-y$,", "$$", "\\rho(x,y)=N(x-y)\\le N(x-z)+N(z-y)=\\rho(x,z)+\\rho(z,y).", "$$", "\\vskip-2em" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 220, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.5", "categories": [], "title": "", "contents": [ "If $x$ and $y$ are vectors in a normed vector space $(A,N),$ then", "\\begin{equation} \\label{eq:8.1.2}", "|N(x)-N(y)|\\le N(x-y).", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Since", "$$", "x=y+(x-y),", "$$", "Definition~\\ref{thmtype:8.1.3}\\part{c} with $u=y$ and $v=x-y$ implies that", "$$", "N(x)\\le N(y)+N(x-y),", "$$", "or", "$$", "N(x)-N(y)\\le N(x-y).", "$$", "Interchanging $x$ and $y$ yields", "$$", "N(y)-N(x)\\le N(y-x).", "$$", "Since $N(x-y)=N(y-x)$ (Definition~\\ref{thmtype:8.1.3}\\part{b} with", "$u=x-y$ and $a=-1$), the last two inequalities imply \\eqref{eq:8.1.2}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:8.1.3", "TRENCH_REAL_ANALYSIS-thmtype:8.1.3" ], "ref_ids": [ 368, 368 ] } ], "ref_ids": [] }, { "id": 221, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.9", "categories": [], "title": "", "contents": [ "If $\\mathbf{X}\\in\\R^n$ and $p_2>p_1\\ge1,$ then", "\\begin{equation} \\label{eq:8.1.12}", "\\|\\mathbf{X}\\|_{p_2}\\le\\|\\mathbf{X}\\|_{p_1};", "\\end{equation}", "moreover,", "\\begin{equation} \\label{eq:8.1.13}", "\\lim_{p\\to\\infty}\\|\\mathbf{X}\\|_{p}=\\max\\set{|x_i|}{1\\le i\\le n}.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Let $u_1$, $u_2$, \\dots, $u_n$ be", "nonnegative and $M=\\max\\set{u_i}{1\\le i\\le n}$. Define", "$$", "\\sigma(p)=\\left(\\sum_{i=1}^n u_i^p\\right)^{1/p}.", "$$", "Since $u_i/\\sigma(p)\\le1$ and $p_2>p_1$,", "$$", "\\left(\\frac{u_i}{\\sigma(p_2)}\\right)^{p_1}\\ge", "\\left(\\frac{u_i}{\\sigma(p_2)}\\right)^{p_2};", "$$", " therefore,", "$$", "\\frac{\\sigma(p_1)}{\\sigma(p_2)}", "=\\left(\\sum_{i=1}^n\\left(\\frac{", "u_i}{\\sigma(p_2)}\\right)^{p_1}\\right)^{1/p_1}", "\\ge\\left(\\sum_{i=1}^n\\left(\\frac{", "u_i}{\\sigma(p_2)}\\right)^{p_2}\\right)^{1/p_1}=1,", "$$", "so $\\sigma(p_1)\\ge\\sigma(p_2)$.", "Since $M\\le\\sigma(p)\\le Mn^{1/p}$,", "$\\lim_{p\\to\\infty}\\sigma(p)= M$.", "Letting $u_i=|x_i|$ yields \\eqref{eq:8.1.12} and \\eqref{eq:8.1.13}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 222, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.11", "categories": [], "title": "", "contents": [ "\\begin{alist}", "\\item % (a)", " The union of open sets is open.", "\\item % (b)", " The intersection of closed sets is closed.", "\\end{alist}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 223, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.13", "categories": [], "title": "", "contents": [ "contains all its limit points$.$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 224, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.15", "categories": [], "title": "", "contents": [ "\\begin{alist}", "\\item % (a)", "The limit of a convergent sequence is unique$.$", "\\item % (b)", "If $\\lim_{n\\to\\infty}u_n=u,$ then every subsequence of", "$\\{u_n\\}$ converges to $u.$", "\\end{alist}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 225, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.17", "categories": [], "title": "", "contents": [ "If a sequence $\\{u_n\\}$ in a metric space $(A,\\rho)$ is convergent$,$", "then it is a Cauchy sequence." ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\lim_{n\\to\\infty}u_n=u$. If $\\epsilon>0$, there is an integer", "$N$ such that", "$\\rho(u_n,u)<\\epsilon/2$ if $n>N$. Therefore, if $m$, $n>N$, then", "$$", "\\rho(u_n,u_m)\\le\\rho(u_n,u)+\\rho(u,u_m)<\\epsilon.", "$$", "\\vskip-2em" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 226, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.19", "categories": [], "title": "The Principle of Nested Sets", "contents": [ "A metric space $(A,\\rho)$ is complete if and only if every", "nested sequence", "$\\{T_n\\}$ of nonempty closed subsets of $A$ such that", " $\\lim_{n\\to\\infty}d(T_n)=0$", "has a nonempty intersection$.$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $(A,\\rho)$ is complete and $\\{T_n\\}$", "is a nested sequence", " of nonempty closed subsets of $A$ such that", " $\\lim_{n\\to\\infty}d(T_n)=0$.", "For each $n$, choose", " $t_n\\in T_n$. If $m\\ge n$,", "then $t_m$, $t_n\\in T_n$, so $\\rho(t_n,t_m)1$ and we have specified $n_1$, $n_2$, \\dots, $n_{j-1}$", "and", "$T_1$, $T_2$, \\dots, $T_{j-1}$. Choose $n_j>n_{j-1}$ so that", "$\\rho(t_n,t_{n_j})<2^{-j}$ if $n\\ge n_j$, and let", "$T_j=\\set{t}{\\rho(t,t_{n_j})\\le2^{-j+1}}$. Then $T_j$ is closed", "and nonempty, $T_{j+1}\\subset T_j$ for all $j$, and", "$\\lim_{j\\to\\infty}d(T_j)=0$. Moreover, $t_n\\in T_j$ if $n\\ge n_j$.", "Therefore, if $\\overline t\\in\\cap_{j=1}^\\infty T_j$, then", "$\\rho(t_n,\\overline t)<2^{-j}$, $n\\ge n_j$, so", "$\\lim_{n\\to\\infty}t_n=\\overline t$, contrary to our assumption.", "Hence, $\\cap_{j=1}^\\infty T_j=\\emptyset$.", "\\boxit{Equivalent Metrics}", "When considering more than one metric on a given set $A$", "we must be careful, for example, in saying that a set is open,", "or that a sequence converges, etc., since the truth or falsity", "of the statement will in general depend on the metric as well as the", "set on which it is imposed. In this situation we will alway", "refer to the metric space by its ``full name;\" that is, $(A,\\rho)$", "rather than just $A$.", "\\begin{definition} \\label{thmtype:8.1.20}", "If $\\rho$ and $\\sigma$ are both metrics on a set $A$, then $\\rho$", "and $\\sigma$ are {\\it equivalent \\/}", "\\hskip-.2em if there are positive constants $\\alpha$ and $\\beta$", "such that", "\\begin{equation} \\label{eq:8.1.18}", "\\alpha\\le\\frac{\\rho(x,y)}{\\sigma(x,y)}\\le\\beta", "\\mbox{\\quad for all \\quad}x,y\\in A\\mbox{\\quad such that \\quad}x\\ne y.", "\\end{equation}", "\\end{definition}" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 227, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.21", "categories": [], "title": "", "contents": [ "If $\\rho$ and $\\sigma$ are equivalent metrics on a set $A,$ then", " $(A,\\rho)$ and $(A,\\sigma)$ have the same open sets." ], "refs": [], "proofs": [ { "contents": [ "Suppose that \\eqref{eq:8.1.18} holds. Let $S$ be an open set in", "$(A,\\rho)$ and let $x_0\\in S$. Then there is an $\\epsilon>0$ such", "that $x\\in S$ if $\\rho(x,x_0)<\\epsilon$, so the second", "inequality in \\eqref{eq:8.1.18}", "implies that $x_0\\in S$ if $\\sigma(x,x_0)\\le\\epsilon/\\beta$.", "Therefore, $S$ is open in $(A,\\sigma)$.", "Conversely, suppose that $S$ is open in $(A,\\sigma)$", "and let $x_0\\in S$. Then there is an $\\epsilon>0$ such", "that $x\\in S$ if $\\sigma(x,x_0)<\\epsilon$, so the first", "inequality in \\eqref{eq:8.1.18}", "implies that $x_0\\in S$ if $\\rho(x,x_0)\\le\\epsilon\\alpha$.", "Therefore, $S$ is open in $(A,\\rho)$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 228, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.22", "categories": [], "title": "", "contents": [ "Any two norms $N_1$ and $N_2$ on $\\R^n$ induce equivalent", "metrics on~$\\R^n.$" ], "refs": [], "proofs": [ { "contents": [ "It suffices to show that there are positive constants $\\alpha$", "and $\\beta$ such", "\\begin{equation} \\label{eq:8.1.19}", "\\alpha\\le\\frac{N_1(\\mathbf{X})}{N_2{(\\bf X})}\\le\\beta\\mbox{\\quad if", "\\quad}", "\\mathbf{X}\\ne\\mathbf{0}.", "\\end{equation}", "We will show that if $N$ is any norm on $\\R^n$, there are", "positive constants $a_N$ and $b_N$ such that", "\\begin{equation} \\label{eq:8.1.20}", "a_N\\|\\mathbf{X}\\|_2\\le N(\\mathbf{X})\\le b_N\\|\\mathbf{X}\\|_2 \\mbox{\\quad if", "\\quad}", "\\mathbf{X}\\ne\\mathbf{0}", "\\end{equation}", "and leave it to you to verify that this implies \\eqref{eq:8.1.19}", "with $\\alpha=a_{N_1}/b_{N_2}$ and $\\beta=b_{N_1}/a_{N_2}$.", "We write $\\mathbf{X}-\\mathbf{Y}=(x_1,x_2, \\dots,x_n)$ as", "$$", "\\mathbf{X}-\\mathbf{Y}=\\sum_{i=1}^n\\,(x_i-y_i)\\mathbf{E}_i,", "$$", "where $\\mathbf{E}_i$ is the vector with $i$th component equal to $1$", "and all other components equal to $0$. From", "Definition~\\ref{thmtype:8.1.3}\\part{b}, \\part{c}, and induction,", "$$", "N(\\mathbf{X}-\\mathbf{Y})\\le\\sum_{i=1}^n|x_i-y_i|N(\\mathbf{E_i});", "$$", "therefore, by Schwarz's inequality,", "\\begin{equation} \\label{eq:8.1.21}", "N(\\mathbf{X}-\\mathbf{Y})\\le K\\|\\mathbf{X}-\\mathbf{Y}\\|_2,", "\\end{equation}", "where", "$$", "K=\\left(\\sum_{i=1}^nN^2(\\mathbf{E_i})\\right)^{1/2}.", "$$", "From \\eqref{eq:8.1.21} and Theorem~\\ref{thmtype:8.1.5},", "$$", "|N(\\mathbf{X})-N(\\mathbf{Y})|\\le K\\|\\mathbf{X}-\\mathbf{Y}\\|_2,", "$$", "so $N$ is continuous on $\\R_2^n=\\R^n$.", "By Theorem~\\ref{thmtype:5.2.12}, there are vectors", "$\\mathbf{U}_1$ and $\\mathbf{U}_2$ such that $\\|\\mathbf{U}_1\\|_2=", "\\|\\mathbf{U}_2\\|_2=1$,", "$$", "N(\\mathbf{U}_1)=\\min\\set{N(\\mathbf{U})}{\\|\\mathbf{U}\\|_2=1},", "\\mbox{\\quad and \\quad}", "N(\\mathbf{U}_2)=\\max\\set{N(\\mathbf{U})}{\\|\\mathbf{U}\\|_2=1}.", "$$", "If", "$a_N=N(\\mathbf{U}_1)$ and $b_N=N(\\mathbf{U}_2)$, then", "$a_N$ and $b_N$ are positive", "(Definition~\\ref{thmtype:8.1.3}\\part{a}), and", "$$", "a_N\\le N\\left(\\frac{\\mathbf{X}}{\\|\\mathbf{X}\\|_2}\\right)\\le b_N", "\\mbox{\\quad if \\quad} \\mathbf{X}\\ne\\mathbf{0}.", "$$", "This and Definition~\\ref{thmtype:8.1.3}\\part{b} imply", "\\eqref{eq:8.1.20}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:8.1.3", "TRENCH_REAL_ANALYSIS-thmtype:8.1.5", "TRENCH_REAL_ANALYSIS-thmtype:5.2.12", "TRENCH_REAL_ANALYSIS-thmtype:8.1.3", "TRENCH_REAL_ANALYSIS-thmtype:8.1.3" ], "ref_ids": [ 368, 220, 152, 368, 368 ] } ], "ref_ids": [] }, { "id": 229, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.23", "categories": [], "title": "", "contents": [ "Suppose that $\\rho$ and $\\sigma$ are equivalent metrics on $A.$ Then", "\\begin{alist}", "\\item % (a)", "A sequence $\\{u_n\\}$ converges to $u$ in $(A,\\rho)$ if and only", "if it converges to $u$ in~$(A,\\sigma).$", "\\item % (a)", "A sequence $\\{u_n\\}$ is a Cauchy sequence in $(A,\\rho)$ if and only", "if it is a Cauchy sequence in $(A,\\sigma).$", "\\item % (b)", "$(A,\\rho)$ is complete if and only if $(A,\\sigma)$ is complete$.$", "\\end{alist}" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 230, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.3", "categories": [], "title": "", "contents": [ "An infinite subset $T$ of $A$ is compact", "if and only if every infinite subset of $T$ has a limit point in $T.$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $T$ has an infinite", "subset $E$ with no limit point in $T$. Then, if $t\\in T$,", " there is an open set $H_t$ such that $t\\in H_t$ and $H_t$", "contains at most one member of $E$. Then ${\\mathcal", "H}=\\cup\\set{H_t}{t\\in T}$ is an open covering of $T$, but", " no finite collection $\\{H_{t_1},H_{t_2}, \\dots,H_{t_k}\\}$ of sets", "from ${\\mathcal H}$ can cover $E$, since $E$ is infinite. Therefore, no", "such collection can cover $T$;", "that is, $T$ is not compact.", "Now suppose that every infinite subset of $T$ has a limit point in", "$T$, and let", "${\\mathcal H}$ be an open covering of $T$.", "We first show that there is a sequence", "$\\{H_i\\}_{i=1}^\\infty$ of sets from ${\\mathcal H}$ that covers $T$.", "If $\\epsilon>0$, then $T$ can be covered by", " $\\epsilon$-neighborhoods of finitely many points of $T$.", "We prove this by contradiction.", "Let $t_1\\in T$. If", "$N_\\epsilon(t_1)$ does not cover $T$, there is a $t_2\\in T$ such", "that", "$\\rho(t_1,t_2)\\ge\\epsilon$.", "Now suppose that $n\\ge 2$ and we have chosen $t_1$, $t_2$, \\dots, $t_n$", "such that $\\rho(t_i,t_j)\\ge\\epsilon$, $1\\le i0$ such that", "$N_\\epsilon(t)\\subset H$. Since $t\\in G_j$ for infinitely", "many values of $j$ and $\\lim_{j\\to\\infty}d(G_j)=0$,", "$$", "G_j\\subset N_\\epsilon(t)\\subset H", "$$", "for some $j$. Therefore,", "if $\\{G_{j_i}\\}_{i=1}^\\infty$", "is the subsequence of $\\{G_j\\}$ such that $G_{j_i}$ is a subset of", "some $H_i$ in ${\\mathcal H}$ (the $\\{H_i\\}$ are not", "necessarily distinct), then", "\\begin{equation} \\label{eq:8.2.1}", "T\\subset\\bigcup_{i=1}^\\infty H_i.", "\\end{equation}", "We will now show that", "\\begin{equation} \\label{eq:8.2.2}", "T\\subset \\bigcup_{i=1}^N H_i.", "\\end{equation}", "for some integer $N$. If this is not so, there is an infinite", "sequence $\\{t_n\\}_{n=1}^\\infty$ in $T$ such that", "\\begin{equation} \\label{eq:8.2.3}", "t_n\\notin \\bigcup_{i=1}^n H_i, \\quad n\\ge 1.", "\\end{equation}", "From our assumption,", " $\\{t_n\\}_{n=1}^\\infty$", "has a limit $\\overline t$ in $T$. From \\eqref{eq:8.2.1},", "$\\overline t\\in H_k$ for some $k$, so", "$N_\\epsilon(\\overline t)\\subset H_k$ for some $\\epsilon>0$. Since", "$\\lim_{n\\to\\infty}t_n=\\overline t$, there is an integer $N$ such that", "$$", "t_n\\in N_\\epsilon(\\overline t)\\subset H_k\\subset \\bigcup_{i=1}^nH_i,\\quad", "n>k,", "$$", "which contradicts \\eqref{eq:8.2.3}. This verifies \\eqref{eq:8.2.2},", "so $T$ is compact." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 231, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.4", "categories": [], "title": "", "contents": [ "A subset $T$ of a metric $A$ is compact if and only if", "every infinite sequence $\\{t_n\\}$ of members of $T$ has a", "subsequence that converges to a member of $T.$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $T$ is compact and $\\{t_n\\}\\subset T$. If $\\{t_n\\}$", "has only finitely many distinct terms, there is a $\\overline t$", "in $T$ such that $t_n=\\overline t$ for infinitely many values of $n$;", "if this is so for $n_10$. Since $\\{t_n\\}$ is a Cauchy sequence,", "there is an integer $N$ such that $\\rho(t_n,t_m)<\\epsilon$,", " $n>m\\ge N$. From \\eqref{eq:8.2.4},", "there is an $m=n_j\\ge N$ such that $\\rho(t_m,\\overline t)<\\epsilon$.", "Therefore,", "$$", "\\rho(t_n,\\overline t)\\le \\rho(t_n,t_m)+\\rho(t_m,\\overline", "t)<2\\epsilon,\\quad n\\ge m.", "$$", "\\vskip-2em" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:8.2.4" ], "ref_ids": [ 231 ] } ], "ref_ids": [] }, { "id": 233, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.6", "categories": [], "title": "", "contents": [ "If $T$ is", "compact$,$ then $T$ is closed and bounded." ], "refs": [], "proofs": [ { "contents": [ "Suppose that", " $\\overline t$ is a limit point of $T$. For each $n$, choose", "$t_n\\ne\\overline t\\in", "B_{1/n}(\\overline t)\\cap T$. Then $\\lim_{n\\to\\infty}t_n=\\overline t$.", "Since every subsequence of $\\{t_n\\}$ also converges to $\\overline t$,", " $\\overline t\\in T$, by", "Theorem~\\ref{thmtype:8.2.3}. Therefore, $T$ is closed.", "The family of unit open balls", "${\\mathcal H}=\\set{B_1(t)}{t\\in T}$", "is an open covering of $T$. Since $T$ is compact, there are", "finitely many members $t_1$, $t_2$, \\dots, $t_n$ of $T$ such that", "$S\\subset \\cup_{j=1}^nB_1(t_j)$. If $u$ and $v$ are arbitrary", "members of $T$, then $u\\in B_1(t_r)$ and $v\\in B_1(t_s)$ for some", "$r$ and $s$ in $\\{1,2, \\dots,n\\}$, so", "\\begin{eqnarray*}", "\\rho(u,v)\\ar\\le \\rho(u,t_r)+\\rho(t_r,t_s)+\\rho(t_s,v)\\\\", "\\ar\\le 2+\\rho(t_r,t_s)\\le2+\\max\\set{\\rho(t_i,t_j)}{1\\le i0$", "such that there is no finite $\\epsilon$-net for $T$.", "Let $t_1\\in T$. Then there must be a $t_2$ in $T$", "such that $\\rho(t_1,t_2)>\\epsilon$. (If not, the singleton", "set $\\{t_1\\}$ would be a finite $\\epsilon$-net for $T$.)", "Now suppose that $n\\ge 2$ and we have chosen $t_1$, $t_2$, \\dots, $t_n$", "such that $\\rho(t_i,t_j)\\ge\\epsilon$, $1\\le i1$ and we have chosen", "an infinite subsequence $\\{s_{i,n-1}\\}_{i=1}^\\infty$ of", "$\\{s_{i,n-2}\\}_{i=1}^\\infty$.", "Since $T_{1/n}$ is finite and $\\{s_{i,n-1}\\}_{i=1}^\\infty$", "is infinite,", "there must be member $t_n$ of $T_{1/n}$ such that", "$\\rho(s_{i,n-1},t_n)\\le1/n$ for infinitely many values of $i$.", "Let $\\{s_{in}\\}_{i=1}^\\infty$ be the subsequence of", "$\\{s_{i,n-1}\\}_{i=1}^\\infty$ such that $\\rho(s_{in},t_n)\\le1/n$.", "From the triangle inequality,", "\\begin{equation} \\label{eq:8.2.5}", "\\rho(s_{in},s_{jn})\\le2/n,\\quad i,j\\ge1,\\quad n\\ge 1.", "\\end{equation}", "Now let $\\widehat s_i=s_{ii}$, $i\\ge 1$. Then $\\{\\widehat s_i\\}_{i=1}^\\infty$", "is an infinite sequence of members of $T$. Moroever, if", "$i,j\\ge n$, then $\\widehat s_i$ and $\\widehat s_j$ are both included in", "$\\{s_{in}\\}_{i=1}^\\infty$, so \\eqref{eq:8.2.5} implies that", "$\\rho(\\widehat s_i,\\widehat s_j)\\le2/n$; that is, $\\{\\widehat s_i\\}_{i=1}^\\infty$", "is a Cauchy sequence and therefore has a limit, since $(A,\\rho)$", " is complete." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:8.2.4" ], "ref_ids": [ 231 ] } ], "ref_ids": [] }, { "id": 236, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.11", "categories": [], "title": "", "contents": [ "A nonempty subset $T$ of $C[a,b]$ is compact if and only if", "it is closed$,$ uniformly bounded$,$ and equicontinuous." ], "refs": [], "proofs": [ { "contents": [ "For necessity, suppose that $T$ is compact. Then $T$ is closed", "(Theorem~\\ref{thmtype:8.2.6}) and totally bounded", "(Theorem~\\ref{thmtype:8.2.8}). Therefore, if $\\epsilon>0$, there is", "a finite subset $T_\\epsilon=\\{g_1,g_2, \\dots,g_k\\}$ of $C[a,b]$", "such that if $f\\in T$, then", "$\\|f-g_i\\|\\le \\epsilon$", "for some $i$ in $\\{1,2, \\dots,k\\}$.", "If we temporarily let $\\epsilon=1$, this implies that", "$$", "\\|f\\|=\\|(f-g_i)+g_i\\|\\le\\|f-g_i\\|+\\|g_i\\|\\le 1+\\|g_i\\|,", "$$", "which implies \\eqref{eq:8.2.6} with", "$$", "M=1+\\max\\set{\\|g_i\\|}{1\\le i\\le k}.", "$$", "For \\eqref{eq:8.2.7}, we again let $\\epsilon$ be arbitary, and write", "\\begin{equation} \\label{eq:8.2.8}", "\\begin{array}{rcl}", "|f(x_1)-f(x_2)|", "\\ar\\le |f(x_1)-g_i(x_1)|+|g_i(x_1)-g_i(x_2)|+|g_i(x_2)-f(x_2)|\\\\", "\\ar\\le |g_i(x_1)-g_i(x_2)|+2\\|f-g_i\\|\\\\", "\\ar< |g_i(x_1)-g_i(x_2)|+2\\epsilon.", "\\end{array}", "\\end{equation}", "Since each of the finitely many functions $g_1$, $g_2$, \\dots, $g_k$", "is uniformly continuous on $[a,b]$", "(Theorem~\\ref{thmtype:2.2.12}), there is a $\\delta>0$ such that", "$$", "|g_i(x_1)-g_i(x_2)|<\\epsilon\\mbox{\\quad if \\quad}", "|x_1-x_2|<\\delta,\\quad 1\\le i\\le k.", "$$", "This and \\eqref{eq:8.2.8} imply \\eqref{eq:8.2.7} with $\\epsilon$", "replaced by $3\\epsilon$. Since this replacement is of no consequence,", "this proves necessity.", "For sufficiency, we will show that $T$ is totally bounded.", " Since $T$ is closed by assumption and", "$C[a,b]$ is complete, Theorem~\\ref{thmtype:8.2.9} will then imply that", "$T$ is compact.", "Let $m$ and $n$ be positive integers and let", "$$", "\\xi_r=a+\\frac{r}{m}(b-a),\\quad 0\\le r\\le m,", "\\mbox{\\quad and \\quad}", "\\eta_s=\\frac{sM}{n},\\quad -n\\le s\\le n;", "$$", "that is, $a=\\xi_0<\\xi_1<\\cdots<\\xi_m=b$ is a partition of $[a,b]$", "into subintervals of length $(b-a)/m$, and", "$-M=\\eta_{-n}<\\eta_{-n+1}<\\cdots<\\eta_{n-1}<\\eta_n=M$ is a partition", "of the \\phantom{segment}", "\\newpage", "\\noindent", " segment of the $y$-axis", "between $y=-M$ and $y=M$ into", "subsegments of length $M/n$.", "Let $S_{mn}$ be the subset of $C[a,b]$ consisting of functions $g$", "such that", "$$", "\\{g(\\xi_0), g(\\xi_1), \\dots, g(\\xi_m)\\}", "\\subset\\{\\eta_{-n},\\eta_{-n+1} \\dots,\\eta_{n-1}, \\eta_n\\}", "$$", " and $g$ is linear on", " $[\\xi_{i-1},\\xi_i]$,", "$1\\le i\\le m$.", " Since there are only $(m+1)(2n+1)$", "points", "of the form $(\\xi_r,\\eta_s)$, $S_{mn}$ is a finite subset of", "$C[a,b]$.", "Now suppose that $\\epsilon>0$, and choose $\\delta>0$ to satisfy", "\\eqref{eq:8.2.7}. Choose $m$ and $n$ so that $(b-a)/m<\\delta$", "and $2M/n<\\epsilon$. If $f$ is an arbitrary member of $T$,", "there is a $g$ in $S_{mn}$ such that", "\\begin{equation} \\label{eq:8.2.9}", "|g(\\xi_i)-f(\\xi_i)|<\\epsilon,\\quad", "0\\le i\\le m.", "\\end{equation}", "If $0\\le i\\le m-1$,", "\\begin{equation} \\label{eq:8.2.10}", "|g(\\xi_i)-g(\\xi_{i+1})|=|g(\\xi_i)-f(\\xi_i)|+|f(\\xi_i)-f(\\xi_{i+1})|", "+|f(\\xi_{i+1})-g(\\xi_{i+1})|.", "\\end{equation}", "Since $\\xi_{i+1}-\\xi_i<\\delta$, \\eqref{eq:8.2.7}, \\eqref{eq:8.2.9},", "and \\eqref{eq:8.2.10} imply that", "$$", "|g(\\xi_i)-g(\\xi_{i+1})|<3\\epsilon.", "$$", "Therefore,", "\\begin{equation} \\label{eq:8.2.11}", "|g(\\xi_i)-g(x)|<3\\epsilon,\\quad \\xi_i\\le x\\le \\xi_{i+1},", "\\end{equation}", "since $g$ is linear on $[\\xi_i,\\xi_{i+1}]$.", "Now let $x$ be an arbitrary point in $[a,b]$, and choose $i$", "so that $x\\in[\\xi_i,\\xi_{i+1}]$. Then", "$$", "|f(x)-g(x)|\\le|f(x)-f(\\xi_i)|+|f(\\xi_i)-g(\\xi_i)|+|g(\\xi_i)-g(x)|,", "$$", "so \\eqref{eq:8.2.7}, \\eqref{eq:8.2.9}, and \\eqref{eq:8.2.11} imply that", "$|f(x)-g(x)|<5\\epsilon$, $a\\le x\\le b$. Therefore,", "$S_{mn}$ is a finite $5\\epsilon$-net for $T$, so $T$ is totally", "bounded." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:8.2.6", "TRENCH_REAL_ANALYSIS-thmtype:8.2.8", "TRENCH_REAL_ANALYSIS-thmtype:2.2.12", "TRENCH_REAL_ANALYSIS-thmtype:8.2.9" ], "ref_ids": [ 233, 234, 25, 235 ] } ], "ref_ids": [] }, { "id": 237, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.12", "categories": [], "title": "", "contents": [ "Suppose that ${\\mathcal F}$ is an infinite uniformly bounded and equicontinuous", "family of functions on $[a,b].$ Then there is a sequence $\\{f_n\\}$", "in ${\\mathcal F}$ that converges uniformly to a continuous function", " on $[a,b].$" ], "refs": [], "proofs": [ { "contents": [ "Let $T$ be the closure of ${\\mathcal F}$; that is, $f\\in T$", "if and only if either $f\\in T$ or $f$ is the uniform limit", "of a sequence of members of ${\\mathcal F}$. Then $T$ is also", "uniformly bounded and equicontinuous (verify),", "and $T$ is closed. Hence, $T$ is compact, by", "Theorem~\\ref{thmtype:8.2.12}. Therefore, ${\\mathcal F}$ has a limit point", "in $T$. (In this context, the limit point is a function $f$ in", "$T$.) Since $f$ is a limit point of ${\\mathcal F}$, there is for each", "integer $n$ a function $f_n$ in ${\\mathcal F}$ such that $\\|f_n-f\\|<1/n$;", "that is $\\{f_n\\}$ converges uniformly to $f$ on $[a,b]$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:8.2.12" ], "ref_ids": [ 237 ] } ], "ref_ids": [] }, { "id": 238, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.3", "categories": [], "title": "", "contents": [ "Suppose that $\\widehat u\\in\\overline D_f.$ Then", "\\begin{equation} \\label{eq:8.3.3}", "\\lim_{u\\to \\widehat u}f(u)=\\widehat v", "\\end{equation}", "if and only if", "\\begin{equation} \\label{eq:8.3.4}", "\\lim_{n\\to\\infty}f(u_n)=\\widehat v", "\\end{equation}", "for every sequence $\\{u_n\\}$ in $D_f$ such that", "\\begin{equation} \\label{eq:8.3.5}", "\\lim_{n\\to\\infty}u_n=\\widehat u.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Suppose that \\eqref{eq:8.3.3} is true, and let $\\{u_n\\}$ be a sequence in", "$D_f$ that satisfies \\eqref{eq:8.3.5}. Let $\\epsilon>0$ and choose", "$\\delta>0$ to satisfy \\eqref{eq:8.3.1}. From \\eqref{eq:8.3.5}, there is", "an integer $N$ such that $\\rho(u_n,\\widehat u)<\\delta$ if $n\\ge N$.", "Therefore, $\\sigma(f(u_n),\\widehat v)<\\epsilon$ if $n\\ge N$, which implies", "\\eqref{eq:8.3.4}.", "For the converse, suppose that \\eqref{eq:8.3.3} is false.", "Then there is an $\\epsilon_0>0$ and a sequence $\\{u_n\\}$", "in $D_f$ such that $\\rho(u_n,\\widehat u)<1/n$ and $\\sigma(f(u_n),\\widehat", "v)\\ge\\epsilon_0$, so \\eqref{eq:8.3.4} is false.", "\\mbox{}\\hfill" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 239, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.4", "categories": [], "title": "", "contents": [ "A function $f$ is continuous at $\\widehat u$ if and", "only if", "$$", "\\lim_{u\\to\\widehat u} f(u)=f(\\widehat u).", "$$" ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 240, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.5", "categories": [], "title": "", "contents": [ "A function $f$ is continuous at $\\widehat u$ if and", "only if", "$$", "\\lim_{n\\to\\infty} f(u_n)=f(\\widehat u)", "$$", "whenever $\\{u_n\\}$ is a sequence in $D_f$ that converges to $\\widehat", "u$." ], "refs": [], "proofs": [], "ref_ids": [] }, { "id": 241, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.6", "categories": [], "title": "", "contents": [ "If $f$ is continuous on a compact set $T,$ then $f(T)$ is compact." ], "refs": [], "proofs": [ { "contents": [ "Let $\\{v_n\\}$ be an infinite sequence in $f(T)$.", "For each $n$, $v_n=f(u_n)$ for some $u_n\\in T$. Since $T$", "is compact, $\\{u_n\\}$ has a subsequence", "$\\{u_{n_j}\\}$ such that $\\lim_{j\\to\\infty}u_{n_j}=\\widehat u\\in T$", "(Theorem~\\ref{thmtype:8.2.4}).", "From Theorem~\\ref{thmtype:8.3.5},", "$\\lim_{j\\to\\infty}f(u_{n_j})=f(\\widehat", "u)$; that is, $\\lim_{j\\to\\infty}v_{n_j}=f(\\widehat u)$. Therefore, $f(T)$", "is compact, again by", "Theorem~\\ref{thmtype:8.2.4}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:8.2.4", "TRENCH_REAL_ANALYSIS-thmtype:8.3.5", "TRENCH_REAL_ANALYSIS-thmtype:8.2.4" ], "ref_ids": [ 231, 240, 231 ] } ], "ref_ids": [] }, { "id": 242, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.8", "categories": [], "title": "", "contents": [ "If $f$ is continuous on a compact set $T,$", "then $f$ is uniformly continuous on $T$." ], "refs": [], "proofs": [ { "contents": [ "If $f$ is not uniformly continuous on $T$,", "then for some", "$\\epsilon_0>0$", "there are sequences $\\{u_n\\}$ and $\\{v_n\\}$ in $T$ such that", "$\\rho(u_n,v_n)<1/n$ and", "\\begin{equation} \\label{eq:8.3.6}", "\\sigma(f(u_n),f(v_n))\\ge\\epsilon_0.", "\\end{equation}", "Since $T$ is compact,", " $\\{u_n\\}$ has a subsequence", "$\\{u_{n_k}\\}$ that converges to a limit $\\widehat u$ in", "$T$ (Theorem~\\ref{thmtype:8.2.4}). Since", "$\\rho(u_{n_k},v_{n_k})<1/n_k$,", "$\\lim_{k\\to\\infty}v_{n_k}=\\widehat u$ also.", " Then", "$$", "\\lim_{k\\to\\infty}f(u_{n_k})=\\dst\\lim_{k\\to", "\\infty}f(v_{n_k})=f(\\widehat u)", "$$", " (Theorem~~\\ref{thmtype:8.3.5}), which", "contradicts \\eqref{eq:8.3.6}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:8.2.4", "TRENCH_REAL_ANALYSIS-thmtype:8.3.5" ], "ref_ids": [ 231, 240 ] } ], "ref_ids": [] }, { "id": 243, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.10", "categories": [], "title": "Contraction Mapping Theorem", "contents": [ "If $f$ is a contraction of a complete metric space $(A,\\rho),$", "then the equation", "\\begin{equation} \\label{eq:8.3.8}", "f(u)=u", "\\end{equation}", "has a unique solution$.$" ], "refs": [], "proofs": [ { "contents": [ "To see that \\eqref{eq:8.3.8} cannot have more than one solution,", "suppose that $u=f(u)$ and $v=f(v)$. Then", "\\begin{equation} \\label{eq:8.3.9}", "\\rho(u,v)=\\rho(f(u),f(v)).", "\\end{equation}", "However, \\eqref{eq:8.3.7} implies that", "\\begin{equation} \\label{eq:8.3.10}", "\\rho(f(u),f(v))\\le\\alpha\\rho(u,v).", "\\end{equation}", "Since \\eqref{eq:8.3.9} and \\eqref{eq:8.3.10} imply that", "$$", "\\rho(u,v)\\le\\alpha\\rho(u,v)", "$$", "and $\\alpha<1$, it follows that $\\rho(u,v)=0$. Hence $u=v$.", "We will now show that \\eqref{eq:8.3.8} has a solution.", "With $u_0$ arbitrary, define", "\\begin{equation}\\label{eq:8.3.11}", "u_n=f(u_{n-1}),\\quad n\\ge1.", "\\end{equation}", "We will show that $\\{u_n\\}$ converges. From \\eqref{eq:8.3.7} and", "\\eqref{eq:8.3.11},", "\\begin{equation} \\label{eq:8.3.12}", "\\rho(u_{n+1},u_n)=\\rho(f(u_n),f(u_{n-1}))\\le\\alpha\\rho(u_n,u_{n-1}).", "\\end{equation}", "\\newpage", "\\noindent", "The inequality", "\\begin{equation}\\label{eq:8.3.13}", "\\rho(u_{n+1},u_n)\\le \\alpha^n \\rho(u_1,u_0),\\quad n\\ge0,", "\\end{equation}", "follows by induction from \\eqref{eq:8.3.12}. If $n>m$, repeated", "application of the triangle inequality yields", "$$", "\\rho(u_n,u_m)", "\\le", "\\rho(u_n,u_{n-1})+\\rho(u_{n-1},u_{n-2})+\\cdots+\\rho(u_{m+1},u_m),", "$$", "and \\eqref{eq:8.3.13} yields", "$$", "\\rho(u_n,u_m)\\le\\rho(u_1,u_0)\\alpha^m(1+\\alpha+\\cdots+\\alpha^{n-m-1})<", "\\frac{\\alpha^m}{1-\\alpha}.", "$$", "Now it follows that", "$$", "\\rho(u_n,u_m)<\\frac{\\rho(u_1,u_0)}{1-\\alpha}\\alpha^N\\mbox{\\quad", "if\\quad} n,m>N,", "$$", "and, since $\\lim_{N\\to\\infty} \\alpha^N=0$, $\\{u_n\\}$ is a Cauchy", "sequence. Since $A$ is complete, $\\{u_n\\}$ has a limit $\\widehat", "u$. Since $f$ is continuous at", "$\\widehat u$,", "$$", "f(\\widehat u)=\\lim_{n\\to\\infty}f(u_{n-1})=\\lim_{n\\to\\infty}u_n=\\widehat u,", "$$", "where Theorem~~\\ref{thmtype:8.3.5} implies the first equality and", "\\eqref{eq:8.3.11} implies the second." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:8.3.5" ], "ref_ids": [ 240 ] } ], "ref_ids": [] }, { "id": 244, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.2", "categories": [], "title": "", "contents": [ "If $f$ is differentiable at $x_0,$ then", "\\begin{equation}\\label{eq:2.3.3}", "f(x)=f(x_0)+[f'(x_0)+E(x)](x-x_0),", "\\end{equation}", "where $E$ is defined on a neighborhood of $x_0$ and", "$$", "\\lim_{x\\to x_0} E(x)=E(x_0)=0.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Define", "\\begin{equation} \\label{eq:2.3.4}", "E(x)=\\left\\{\\casespace\\begin{array}{ll}", "\\dst\\frac{f(x)-f(x_0)}{ x-x_0}-", "f'(x_0),&x\\in D_f\\mbox{ and }x\\ne x_0,\\\\[2\\jot]", "0,&x=x_0.", "\\end{array}\\right.", "\\end{equation}", "Solving \\eqref{eq:2.3.4} for $f(x)$ yields \\eqref{eq:2.3.3} if $x\\ne x_0$,", "and \\eqref{eq:2.3.3} is obvious if $x=x_0$.", "Definition~\\ref{thmtype:2.3.1}", "implies that $\\lim_{x\\to x_0}E(x)=0$. We defined $E(x_0)=0$ to make", "$E$ continuous at $x_0$.", "\\mbox{}" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.3.1" ], "ref_ids": [ 313 ] } ], "ref_ids": [] }, { "id": 245, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.5.2", "categories": [], "title": "", "contents": [ "If $f^{(n)}(x_0)$ exists$,$ then", "\\begin{equation}\\label{eq:2.5.7}", "f(x)=\\sum_{r=0}^n\\frac{f^{(r)}(x_0)}{ r!} (x-x_0)^r+E_n(x)(x-x_0)^n,", "\\end{equation}", "where", "$$", "\\lim_{x\\to x_0} E_n(x)=E_n(x_0)=0.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Define", "$$", "E_n(x)=", "\\left\\{\\casespace\\begin{array}{ll}", "\\dst\\frac{f(x)-T_n(x)}{(x-x_0)^n},&x\\in D_f-\\{x_0\\},\\\\", "0,&x=x_0.\\end{array}\\right.", "$$", "Then \\eqref{eq:2.5.5} implies that $\\lim_{x\\to x_0}E_n(x)=E_n(x_0)=0$,", "and it is straightforward to verify \\eqref{eq:2.5.7}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 246, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.1", "categories": [], "title": "", "contents": [ "Suppose that", "\\begin{equation} \\label{eq:3.2.1}", "|f(x)|\\le M,\\quad a\\le x\\le b,", "\\end{equation}", "and let $P'$ be a partition of $[a,b]$ obtained by adding $r$ points to a", "partition $P=\\{x_0,x_1, \\dots,x_n\\}$ of $[a,b].$ Then", "\\begin{eqnarray}", "S(P)\\ge S(P')\\ar\\ge S(P)-2Mr\\|P\\|\\label{eq:3.2.2}\\\\", "\\arraytext{and}\\nonumber\\\\", "s(P)\\le s(P')\\ar\\le s(P)+2Mr\\|P\\|\\label{eq:3.2.3}.", "\\end{eqnarray}" ], "refs": [], "proofs": [ { "contents": [ "We will prove \\eqref{eq:3.2.2} and leave the proof of \\eqref{eq:3.2.3}", "to you (Exercise~\\ref{exer:3.2.1}).", "First suppose that $r=1$, so", " $P'$ is obtained by adding one point $c$ to the", "partition", "$P=\\{x_0,x_1, \\dots,x_n\\}$; then", "$x_{i-1}1$ and $P'$ is obtained by adding points $c_1$,", "$c_2$, \\dots, $c_r$ to $P$. Let $P^{(0)}=P$ and, for $j\\ge1$, let", "$P^{(j)}$ be the partition of $[a,b]$ obtained by adding $c_j$", "to $P^{(j-1)}$. Then the result just proved implies that", "$$", "0\\le S(P^{(j-1)})-S(P^{(j)})\\le2M\\|P^{(j-1)}\\|,\\quad 1\\le j\\le r.", "$$", "\\newpage", "\\noindent", "Adding these inequalities and taking account of cancellations", " yields", "\\begin{equation} \\label{eq:3.2.5}", "0\\le", "S(P^{(0)})-S(P^{(r)})\\le2M(\\|P^{(0)}\\|+\\|P^{(1)}\\|+\\cdots+\\|P^{(r-1)}\\|).", "\\end{equation}", "Since $P^{(0)}=P$, $P^{(r)}=P'$, and $\\|P^{(k)}\\|\\le\\|P^{(k-1)}\\|$", "for $1\\le k\\le r-1$, \\eqref{eq:3.2.5} implies that", "$$", "0\\le S(P)-S(P')\\le 2Mr\\|P\\|,", "$$", "which is equivalent to \\eqref{eq:3.2.2}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 247, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.4", "categories": [], "title": "", "contents": [ "If $f$ is bounded on $[a,b]$ and", " $\\epsilon>0,$ there is a $\\delta>0$ such that", "\\begin{equation} \\label{eq:3.2.12}", "\\overline{\\int_a^b}f(x)\\,dx\\le", "S(P)<\\overline{\\int_a^b}f(x)\\,dx+\\epsilon", "\\end{equation}", "and", "$$", "\\underline{\\int_a^b} f(x)\\,dx\\ge s(P)>\\underline{\\int_a^b}", "f(x)\\,dx-\\epsilon", "$$", "if $\\|P\\|<\\delta$." ], "refs": [], "proofs": [ { "contents": [ "We show that \\eqref{eq:3.2.12} holds if $\\|P\\|$ is sufficiently", "small, and leave the rest of the proof to you (Exercise~\\ref{exer:3.2.3}).", "The first inequality in \\eqref{eq:3.2.12} follows immediately from", "Definition~\\ref{thmtype:3.1.3}.", " To establish the second inequality,", "suppose that $|f(x)|\\le K$ if $a\\le x\\le b$. From", "Definition~\\ref{thmtype:3.1.3}, there is a partition $P_0=", "\\{x_0,x_1, \\dots,x_{r+1}\\}$ of $[a,b]$ such that", "\\begin{equation} \\label{eq:3.2.13}", "S(P_0)<\\overline{\\int_a^b}f(x)\\,dx+\\frac{\\epsilon}{2}.", "\\end{equation}", "If $P$ is any partition of $[a,b]$, let $P'$ be constructed from the", "partition points of $P_0$ and $P$. Then", "\\begin{equation} \\label{eq:3.2.14}", "S(P')\\le S(P_0),", "\\end{equation}", "by Lemma~\\ref{thmtype:3.2.1}. Since $P'$ is obtained by adding at most", "$r$ points to $P$, Lemma~\\ref{thmtype:3.2.1} implies that", "\\begin{equation} \\label{eq:3.2.15}", "S(P')\\ge S(P)-2Kr\\|P\\|.", "\\end{equation}", " Now \\eqref{eq:3.2.13}, \\eqref{eq:3.2.14}, and \\eqref{eq:3.2.15}", "imply that", "\\begin{eqnarray*}", "S(P)\\ar\\le S(P')+2Kr\\|P\\|\\\\", "\\ar\\le S(P_0)+2Kr\\|P\\|\\\\", "&<&\\overline{\\int_a^b} f(x)\\,dx+\\frac{\\epsilon}{2}+2Kr\\|P\\|.", "\\end{eqnarray*}", " Therefore, \\eqref{eq:3.2.12} holds if", "$$", "\\|P\\|<\\delta=\\frac{\\epsilon}{4Kr}.", "$$", "\\vskip-4.5ex" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:3.1.3", "TRENCH_REAL_ANALYSIS-thmtype:3.1.3", "TRENCH_REAL_ANALYSIS-thmtype:3.2.1", "TRENCH_REAL_ANALYSIS-thmtype:3.2.1" ], "ref_ids": [ 316, 316, 246, 246 ] } ], "ref_ids": [] }, { "id": 248, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.5.3", "categories": [], "title": "", "contents": [ "If $w_f(x)<\\epsilon$ for $a\\le x \\le b,$ then there is a $\\delta>0$", "such", "that $W_f[a_1,b_1]\\le\\epsilon,$ provided that $[a_1,b_1]\\subset", "[a,b]$ and", "$b_1-a_1<\\delta.$" ], "refs": [], "proofs": [ { "contents": [ "We use the Heine--Borel theorem (Theorem~\\ref{thmtype:1.3.7}).", "If $w_f(x)<\\epsilon$, there is an $h_x>0$ such that", "\\begin{equation} \\label{eq:3.5.1}", "|f(x')-f(x'')|<\\epsilon", "\\end{equation}", "\\newpage", "\\noindent", "if", "\\begin{equation} \\label{eq:3.5.2}", "x-2h_x0$, there is an $\\overline{x}$ from $E_\\rho$ in", "$(x_0-h,x_0+h)$.", "Since $[\\overline{x}-h_1,\\overline{x}+h_1] \\subset [x_0-h,x_0+h]$ for", "sufficiently small $h_1$ and", " $W_f[\\overline{x}-h_1,\\overline{x}+h_1]\\ge\\rho$, it follows that", " $W_f[x_0-h,x_0+h]\\ge\\rho$ for all", "$h>0$. This implies that $x_0\\in E_\\rho$, so $E_\\rho$ is closed", "(Corollary~\\ref{thmtype:1.3.6}).", "Now we will show that the stated condition in necessary for", "integrability.", "Suppose that the condition is not satisfied; that is, there is a", "$\\rho>0$ and a $\\delta>0$ such that", "$$", "\\sum_{j=1}^p L(I_j)\\ge\\delta", "$$", "\\newpage", "\\noindent", "for every finite set $\\{I_1,I_2, \\dots, I_p\\}$ of open intervals", "covering", "$E_\\rho$. If", "$P=", "\\{x_0,x_1, \\dots,x_n\\}$ is a partition of $[a,b]$, then", "\\begin{equation} \\label{eq:3.5.4}", "S(P)-s(P)=\\sum_{j\\in A} (M_j-m_j)(x_j-x_{j-1})+\\sum_{j\\in B}", "(M_j-m_j)(x_j-x_{j-1}),", "\\end{equation}", "where", "$$", "A=\\set{j}{[x_{j-1},x_j]\\cap E_\\rho\\ne\\emptyset}\\mbox{\\quad", "and\\quad}", "B=\\set{j}{[x_{j-1},x_j]\\cap E_\\rho=\\emptyset}\\negthickspace.", "$$", "Since $\\bigcup_{j\\in A} (x_{j-1},x_j)$ contains all points of $E_\\rho$", "except any of $x_0$, $x_1$, \\dots, $x_n$ that may be in $E_\\rho$, and", "each of", "these finitely many possible exceptions can be covered by an open interval", "of length as small as we please, our assumption on $E_\\rho$ implies that", "$$", "\\sum_{j\\in A} (x_j-x_{j-1})\\ge\\delta.", "$$", "Moreover, if $j\\in A$, then", "$$", "M_j-m_j\\ge\\rho,", "$$", "so \\eqref{eq:3.5.4} implies that", "$$", "S(P)-s(P)\\ge\\rho\\sum_{j\\in A} (x_j-x_{j-1})\\ge\\rho\\delta.", "$$", "Since this holds for every partition of $[a,b]$, $f$ is not integrable on", "$[a,b]$, by Theorem~\\ref{thmtype:3.2.7}. This proves that the stated condition is", "necessary for integrability.", "For sufficiency, let $\\rho$ and $\\delta$ be positive numbers and let", "$I_1$, $I_2$, \\dots, $I_p$ be open intervals that cover $E_\\rho$ and", "satisfy", "\\eqref{eq:3.5.3}. Let", "$$", "\\widetilde{I}_j=[a,b]\\cap\\overline{I}_j.", "$$", "($\\overline{I}_j=\\mbox{closure of } I$.) After combining any of", "$\\widetilde{I}_1$, $\\widetilde{I}_2$, \\dots, $\\widetilde{I}_p$ that overlap, we", "obtain a set of pairwise disjoint closed subintervals", "$$", "C_j=[\\alpha_j,\\beta_j],\\quad 1\\le j\\le q\\ (\\le p),", "$$", "of $[a,b]$ such that", "\\begin{equation} \\label{eq:3.5.5}", "a\\le\\alpha_1<\\beta_1<\\alpha_2<\\beta_2\\cdots<", "\\alpha_{q-1}<\\beta_{q-1}<\\alpha_q<\\beta_q\\le b,", "\\end{equation}", "\\begin{equation} \\label{eq:3.5.6}", "\\sum_{i=1}^q\\, (\\beta_i-\\alpha_i)<\\delta", "\\end{equation}", "and", "$$", "w_f(x)<\\rho,\\quad\\beta_j\\le x\\le\\alpha_{j+1},\\quad 1\\le j\\le q-1.", "$$", "Also, $w_f(x)<\\rho$ for $a\\le x\\le\\alpha_1$ if $a<\\alpha_1$ and for", "$\\beta_q\\le x\\le b$ if $\\beta_q0$, let", "$$", "\\delta=\\frac{\\epsilon}{4K}\\mbox{\\quad", "and\\quad}\\rho=\\frac{\\epsilon}{", "2(b-a)}.", "$$", "Then \\eqref{eq:3.5.7} yields", "$$", "S(P)-s(P)<\\epsilon,", "$$", "and Theorem~\\ref{thmtype:3.2.7} implies that $f$ is", "integrable on $[a,b]$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:1.3.6", "TRENCH_REAL_ANALYSIS-thmtype:3.2.7", "TRENCH_REAL_ANALYSIS-thmtype:3.5.3", "TRENCH_REAL_ANALYSIS-thmtype:3.2.7" ], "ref_ids": [ 274, 50, 248, 50 ] } ], "ref_ids": [] }, { "id": 250, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.4", "categories": [], "title": "", "contents": [ "Suppose that for $n$ sufficiently large", " $($that is$,$ for $n \\ge\\mbox{some", "integer }N$$)$", " the terms of", "$\\sum_{n=k}^\\infty a_n$ satisfy", " some condition that implies convergence", "of an infinite series$.$ Then $\\sum_{n=k}^\\infty a_n$", "converges$.$", "Similarly, suppose that for $n$ sufficiently large the terms", "$\\sum_{n=k}^\\infty a_n$ satisfy", " some condition that implies divergence", "of an infinite series$.$ Then $\\sum_{n=k}^\\infty a_n$", "diverges$.$" ], "refs": [], "proofs": [ { "contents": [ "In terms of the partial sums $\\{A_n\\}$ of $\\sum a_n$,", "$$", "a_n+a_{n+1}+\\cdots+a_m=A_m-A_{n-1}.", "$$", "Therefore, \\eqref{eq:4.3.3} can be written as", "$$", "|A_m-A_{n-1}|<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N.", "$$", "Since $\\sum a_n$ converges if and only if $\\{A_n\\}$ converges,", "Theorem~\\ref{thmtype:4.1.13} implies the conclusion." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.1.13" ], "ref_ids": [ 89 ] } ], "ref_ids": [] }, { "id": 251, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.2", "categories": [], "title": "", "contents": [ "If $g$ and $h$ are defined on $S,$ then", "\\begin{eqnarray*}", "\\|g+h\\|_S\\ar\\le\\|g\\|_S+\\|h\\|_S\\\\", "\\arraytext{and}\\\\", "\\|gh\\|_S\\ar\\le\\|g\\|_S\\|h\\|_S.", "\\end{eqnarray*}", "Moroever$,$ if either $g$ or $h$ is bounded on $S,$ then", "$$", "\\|g-h\\|_S\\ge\\left|\\|g\\|_S-\\|h\\|_S\\|\\right|.", "$$" ], "refs": [], "proofs": [ { "contents": [ "For necessity, suppose that $\\{F_n\\}$ converges uniformly to", "$F$ on $S$. Then, if $\\epsilon>0$, there is an integer $N$ such that", "$$", "\\|F_k-F\\|_S<\\frac{\\epsilon}{2}\\mbox{\\quad if\\quad} k\\ge N.", "$$", "Therefore,", "\\begin{eqnarray*}", "\\|F_n-F_m\\|_S\\ar=\\|(F_n-F)+(F-F_m)\\|_S\\\\", "\\ar\\le \\|F_n-F\\|_S+\\|F-F_m\\|_S \\mbox{\\quad", "(Lemma~\\ref{thmtype:4.4.2})\\quad}\\\\", "&<&\\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}=\\epsilon\\mbox{\\quad if\\quad}", "m, n\\ge N.", "\\end{eqnarray*}", "For sufficiency, we first observe that \\eqref{eq:4.4.2} implies that", "$$", "|F_n(x)-F_m(x)|<\\epsilon\\mbox{\\quad if\\quad} n, m\\ge N,", "$$", "for any fixed $x$ in $S$. Therefore, Cauchy's convergence criterion", "for sequences of constants (Theorem~\\ref{thmtype:4.1.13})", "implies that", "$\\{F_n(x)\\}$ converges for each $x$ in $S$; that is, $\\{F_n\\}$", "converges pointwise to a limit function $F$ on $S$. To see that the", "convergence is uniform, we write", "\\begin{eqnarray*}", "|F_m(x)-F(x) |\\ar=|[F_m(x)-F_n(x)]+[F_n(x)-F(x)]|\\\\", "\\ar\\le |F_m(x)-F_n(x)|+| F_n(x)-F(x)|\\\\", "\\ar\\le \\|F_m-F_n\\|_S+|F_n(x)-F(x)|.", "\\end{eqnarray*}", "This and \\eqref{eq:4.4.2} imply that", "\\begin{equation} \\label{eq:4.4.3}", "|F_m(x)-F(x)|<\\epsilon+|F_n(x)-F(x)|\\quad\\mbox {if}\\quad n, m\\ge N.", "\\end{equation}", "Since $\\lim_{n\\to\\infty}F_n(x)=F(x)$,", "$$", "|F_n(x)-F(x)|<\\epsilon", "$$", "for some $n\\ge N$, so \\eqref{eq:4.4.3} implies that", "$$", "|F_m(x)-F(x)|<2\\epsilon\\mbox{\\quad if\\quad} m\\ge N.", "$$", "But this inequality holds for all $x$ in $S$, so", "$$", "\\|F_m-F\\|_S\\le2\\epsilon\\mbox{\\quad if\\quad} m\\ge N.", "$$", "Since $\\epsilon$ is an arbitrary positive number, this implies that", "$\\{F_n\\}$ converges uniformly to $F$ on~$S$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.4.2", "TRENCH_REAL_ANALYSIS-thmtype:4.1.13" ], "ref_ids": [ 251, 89 ] } ], "ref_ids": [] }, { "id": 252, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.5", "categories": [], "title": "", "contents": [ "If $\\mathbf{X}$ and $\\mathbf{Y}$ are any two vectors in $\\R^n,$ then", "\\begin{equation} \\label{eq:5.1.3}", "|\\mathbf{X}\\cdot\\mathbf{Y}|\\le |\\mathbf{X}|\\,|\\mathbf{Y}|,", "\\end{equation}", "with equality if and only if one of the vectors is a scalar", "multiple of the other$.$" ], "refs": [], "proofs": [ { "contents": [ "If $\\mathbf{Y}=\\mathbf{0}$, then both sides", "of \\eqref{eq:5.1.3} are $\\mathbf{0}$, so \\eqref{eq:5.1.3} holds, with equality.", "In this case, $\\mathbf{Y}=0\\mathbf{X}$.", "Now suppose that $\\mathbf{Y}\\ne\\mathbf{0}$ and", " $t$ is any real number. Then", "\\begin{equation}\\label{eq:5.1.4}", "\\begin{array}{rcl}", "0\\ar\\le \\dst{\\sum^n_{i=1} (x_i-ty_i)^2}\\\\", "\\ar=\\dst{\\sum^n_{i=1} x^2_i-2t\\sum^n_{i=1} x_iy_i+t^2\\sum^n_{i=1}", "y^2_i}\\\\\\\\", "\\ar=|\\mathbf{X}|^2-2(\\mathbf{X}\\cdot\\mathbf{Y})t+t^2|\\mathbf{Y}|^2.", "\\end{array}", "\\end{equation}", "The last expression is a second-degree polynomial $p$", "in $t$. From the quadratic formula, the zeros of $p$ are", "$$", "t=\\frac{(\\mathbf{X}\\cdot\\mathbf{Y})\\pm\\sqrt{(\\mathbf{X}\\cdot\\mathbf{Y})^2-", "|\\mathbf{X}|^2|\\mathbf{Y}|^2}}{ |\\mathbf{Y}|^2}.", "$$", "Hence,", "\\begin{equation}\\label{eq:5.1.5}", "(\\mathbf{X}\\cdot\\mathbf{Y})^2\\le |\\mathbf{X}|^2|\\mathbf{Y}|^2,", "\\end{equation}", "because if not, then $p$ would have two distinct real zeros and", "therefore", "be negative between them (Figure~\\ref{figure:5.1.1}), contradicting the", "inequality \\eqref{eq:5.1.4}. Taking square roots in \\eqref{eq:5.1.5} yields", "\\eqref{eq:5.1.3} if $\\mathbf{Y}\\ne\\mathbf{0}$.", "If $\\mathbf{X}=t\\mathbf{Y}$, then", "$|\\mathbf{X}\\cdot\\mathbf{Y}|=|\\mathbf{X}||\\mathbf{Y}|", "=|t||\\mathbf{Y}|^2$ (verify), so equality holds in \\eqref{eq:5.1.3}.", "Conversely, if equality holds in \\eqref{eq:5.1.3}, then $p$ has the real", "zero $t_0=(\\mathbf{X}\\cdot\\mathbf{Y})/|\\mathbf{Y}\\|^2$, and", "$$", "\\sum_{i=1}^n(x_i-t_0y_i)^2=0", "$$", "\\nopagebreak", "from \\eqref{eq:5.1.4}; therefore, $\\mathbf{X}=t_0\\mathbf{Y}$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 253, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.12", "categories": [], "title": "", "contents": [ "If $\\mathbf{X}_1$ and $\\mathbf{X}_2$ are in $S_r(\\mathbf{X}_0)$ for some $r>0$,", "then so is every point on", "the line segment from $\\mathbf{X}_1$ to $\\mathbf{X}_2.$" ], "refs": [], "proofs": [ { "contents": [ "The line segment is given by", "$$", "\\mathbf{X}=t\\mathbf{X}_2+(1-t)\\mathbf{X}_1,\\quad 00$. If", "$$", "|\\mathbf{X}_1-\\mathbf{X}_0|0$. Our assumptions imply that there is", "a $\\delta>0$ such that $f_{x_1}, f_{x_2}, \\dots, f_{x_n}$ are defined", "in the $n$-ball", "$$", "S_\\delta (\\mathbf{X}_0)=\\set{\\mathbf{X}}{|\\mathbf{X}-\\mathbf{X}_0|<\\delta}", "$$", "and", "\\begin{equation}\\label{eq:5.3.24}", "|f_{x_j}(\\mathbf{X})-f_{x_j}(\\mathbf{X}_0)|<\\epsilon\\mbox{\\quad if\\quad}", "|\\mathbf{X}-\\mathbf{X}_0|<\\delta,\\quad 1\\le j\\le n.", "\\end{equation}", "Let $\\mathbf{X}=(x_1,x_, \\dots,x_n)$ be in $S_\\delta(\\mathbf{X}_0)$.", "Define", "$$", "\\mathbf{X}_j=(x_1, \\dots,x_j, x_{j+1,0}, \\dots,x_{n0}),\\quad 1\\le j\\le n-1,", "$$", "and", "$\\mathbf{X}_n=\\mathbf{X}$.", "Thus, for $1\\le j\\le n$, $\\mathbf{X}_j$ differs from $\\mathbf{X}_{j-1}$", " in the", "$j$th component only, and the line segment from $\\mathbf{X}_{j-1}$ to", "$\\mathbf{X}_j$ is in $S_\\delta (\\mathbf{X}_0)$.", "Now write", "\\begin{equation}\\label{eq:5.3.25}", "f(\\mathbf{X})-f(\\mathbf{X}_0)=f(\\mathbf{X}_n)-f(\\mathbf{X}_0)=", "\\sum^n_{j=1}\\,[f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})],", "\\end{equation}", "and consider the auxiliary functions", "\\begin{equation}\\label{eq:5.3.26}", "\\begin{array}{rcl}", "g_1(t)\\ar=f(t,x_{20}, \\dots,x_{n0}),\\\\[2\\jot]", "g_j(t)\\ar=f(x_1, \\dots,x_{j-1},t,x_{j+1,0}, \\dots,x_{n0}),\\quad 2\\le j\\le", "n-1,\\\\[2\\jot]", "g_n(t)\\ar=f(x_1, \\dots,x_{n-1},t),", "\\end{array}", "\\end{equation}", "where, in each case, all variables except $t$ are temporarily regarded", "as constants. Since", "$$", "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=g_j(x_j)-g_j(x_{j0}),", "$$", "the mean value theorem implies that", "$$", "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=g'_j(\\tau_j)(x_j-x_{j0}),", "$$", "\\newpage", "\\noindent", "where $\\tau_j$ is between $x_j$ and $x_{j0}$. From \\eqref{eq:5.3.26},", "$$", "g'_j(\\tau_j)=f_{x_j}(\\widehat{\\mathbf{X}}_j),", "$$", "where $\\widehat{\\mathbf{X}}_j$ is on the line segment from $\\mathbf{X}_{j-1}$ to", "$\\mathbf{X}_j$. Therefore,", "$$", "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=f_{x_j}(\\widehat{\\mathbf{X}}_j)(x_j-x_{j0}),", "$$", "and \\eqref{eq:5.3.25} implies that", "\\begin{eqnarray*}", "f(\\mathbf{X})-f(\\mathbf{X}_0)\\ar=\\sum^n_{j=1} f_{x_j} (\\widehat{\\mathbf{X}}_j)(x_j-x_{j0})\\\\", "\\ar=\\sum^n_{j=1} f_{x_j}(\\mathbf{X}_0) (x_j-x_{j0})+\\sum^n_{j=1}", "\\,[f_{x_j}(\\widehat{\\mathbf{X}}_j)-f_{x_j}(\\mathbf{X}_0)](x_j-x_{j0}).", "\\end{eqnarray*}", "From this and \\eqref{eq:5.3.24},", "$$", "\\left|f(\\mathbf{X})-f(\\mathbf{X}_0)-\\sum^n_{j=1}", "f_{x_j}(\\mathbf{X}_{0})", "(x_j-x_{j0})\\right|\\le", "\\epsilon\\sum^n_{j=1} |x_j-x_{j0}|\\le n\\epsilon |\\mathbf{X}-\\mathbf{X}_0|,", "$$", "which implies that $f$ is differentiable at $\\mathbf{X}_0$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 255, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.2", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{G}=(g_1,g_2, \\dots,g_n)$ is differentiable at", "$$", " \\mathbf{U}_0=(u_{10}, u_{20}, \\dots,u_{m0}),", "$$", " and", " define", "$$", "M=\\left(\\sum_{i=1}^n\\sum_{j=1}^m\\left(\\frac{\\partial g_i(\\mathbf{U}_0}", "{\\partial u_j}\\right)^2\\right)^{1/2}.", "$$", "Then$,$ if $\\epsilon>0,$ there is a $\\delta>0$ such that", "$$", "\\frac{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}", "{|\\mathbf{U}-\\mathbf{U}_{0}|}", "0,$ there is a $\\delta>0$ such that", "\\begin{equation}\\label{eq:6.2.8}", "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})|<", "(\\|\\mathbf{F}'(\\mathbf{X}_{0})\\|", "+\\epsilon) |\\mathbf{X}-\\mathbf{Y}|", "\\mbox{\\quad if\\quad}\\mathbf{A},\\mathbf{Y}\\in B_\\delta (\\mathbf{X}_0).", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Consider the auxiliary function", "\\begin{equation} \\label{eq:6.2.9}", "\\mathbf{G}(\\mathbf{X})=\\mathbf{F}(\\mathbf{X})-\\mathbf{F}'(\\mathbf{X}_0)\\mathbf{X}.", "\\end{equation}", "The components of $\\mathbf{G}$ are", "$$", "g_i(\\mathbf{X})=f_i(\\mathbf{X})-\\sum_{j=1}^n", "\\frac{\\partial f_i(\\mathbf{X}_{0})", "\\partial x_j} x_j,", "$$", "so", "$$", "\\frac{\\partial g_i(\\mathbf{X})}{\\partial x_j}=", "\\frac{\\partial f_i(\\mathbf{X})}", "{\\partial x_j}-\\frac{\\partial f_i(\\mathbf{X}_0)}{\\partial x_j}.", "$$", "\\newpage", "\\noindent", "Thus, $\\partial g_i/\\partial x_j$ is continuous on $N$ and zero at", "$\\mathbf{X}_0$. Therefore, there is a $\\delta>0$ such that", "\\begin{equation}\\label{eq:6.2.10}", "\\left|\\frac{\\partial g_i(\\mathbf{X})}{\\partial x_j}\\right|<\\frac{\\epsilon}{", "\\sqrt{mn}}\\mbox{\\quad for \\quad}1\\le i\\le m,\\quad 1\\le j\\le n,", "\\mbox{\\quad if \\quad}", "|\\mathbf{X}-\\mathbf{X}_0|<\\delta.", "\\end{equation}", "Now suppose that $\\mathbf{X}$, $\\mathbf{Y}\\in B_\\delta(\\mathbf{X}_0)$. By", "Theorem~\\ref{thmtype:5.4.5},", "\\begin{equation}\\label{eq:6.2.11}", "g_i(\\mathbf{X})-g_i(\\mathbf{Y})=\\sum_{j=1}^n", "\\frac{\\partial g_i(\\mathbf{X}_i)}{\\partial x_j}(x_j-y_j),", "\\end{equation}", "where $\\mathbf{X}_i$ is on the line segment from $\\mathbf{X}$ to $\\mathbf{Y}$,", "so $\\mathbf{X}_i\\in B_\\delta(\\mathbf{X}_0)$. From \\eqref{eq:6.2.10},", "\\eqref{eq:6.2.11}, and Schwarz's inequality,", "$$", "(g_i(\\mathbf{X})-g_i(\\mathbf{Y}))^2\\le\\left(\\sum_{j=1}^n\\left[\\frac{\\partial", "g_i", "(\\mathbf{X}_i)}{\\partial x_j}\\right]^2\\right)", "|\\mathbf{X}-\\mathbf{Y}|^2", "<\\frac{\\epsilon^2}{ m} |\\mathbf{X}-\\mathbf{Y}|^2.", "$$", "Summing this from $i=1$ to $i=m$ and taking square roots yields", "\\begin{equation}\\label{eq:6.2.12}", "|\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})|<\\epsilon", "|\\mathbf{X}-\\mathbf{Y}|", "\\mbox{\\quad if\\quad}\\mathbf{X}, \\mathbf{Y}\\in B_\\delta(\\mathbf{X}_0).", "\\end{equation}", "To complete the proof, we note that", "\\begin{equation}\\label{eq:6.2.13}", "\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})=", "\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})+\\mathbf{F}'(\\mathbf{X}_0)(\\mathbf{X}-\\mathbf{Y}),", "\\end{equation}", " so \\eqref{eq:6.2.12} and the triangle inequality imply \\eqref{eq:6.2.8}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.4.5" ], "ref_ids": [ 164 ] } ], "ref_ids": [] }, { "id": 257, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.6", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{F}:\\R^n\\to\\R^n$ is continuously", "differentiable on a neighborhood of $\\mathbf{X}_0$", " and $\\mathbf{F}'(\\mathbf{X}_0)$ is nonsingular$.$ Let", "\\begin{equation}\\label{eq:6.2.14}", "r=\\frac{1}{\\|(\\mathbf{F}'(\\mathbf{X}_0))^{-1}\\|}.", "\\end{equation}", "Then$,$ for every $\\epsilon>0,$ there is a $\\delta>0$ such that", "\\begin{equation}\\label{eq:6.2.15}", "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})|\\ge (r-\\epsilon)", "|\\mathbf{X}-\\mathbf{Y}|\\mbox{\\quad if\\quad} \\mathbf{X},\\mathbf{Y}\\in", "B_\\delta(\\mathbf{X}_{0}).", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\mathbf{X}$ and $\\mathbf{Y}$ be arbitrary points in", "$D_\\mathbf{F}$ and let $\\mathbf{G}$ be as in \\eqref{eq:6.2.9}. From", "\\eqref{eq:6.2.13},", "\\begin{equation} \\label{eq:6.2.16}", "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})|\\ge\\big|", "|\\mathbf{F}'(\\mathbf{X}_0)(\\mathbf{X}", "-\\mathbf{Y})|-|\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})|\\big|,", "\\end{equation}", "Since", "$$", "\\mathbf{X}-\\mathbf{Y}=[\\mathbf{F}'(\\mathbf{X}_0)]^{-1}", "\\mathbf{F}'(\\mathbf{X}_{0})", "(\\mathbf{X}-\\mathbf{Y}),", "$$", "\\eqref{eq:6.2.14} implies that", "$$", "|\\mathbf{X}-\\mathbf{Y}|\\le \\frac{1}{ r} |\\mathbf{F}'(\\mathbf{X}_0)", "(\\mathbf{X}-\\mathbf{Y}|,", "$$", "so", "\\begin{equation}\\label{eq:6.2.17}", "|\\mathbf{F}'(\\mathbf{X}_0)(\\mathbf{X}-\\mathbf{Y})|\\ge r|\\mathbf{X}-\\mathbf{Y}|.", "\\end{equation}", " Now choose $\\delta>0$ so that \\eqref{eq:6.2.12} holds.", "Then \\eqref{eq:6.2.16} and \\eqref{eq:6.2.17} imply \\eqref{eq:6.2.15}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 258, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.7", "categories": [], "title": "", "contents": [ "If $\\mathbf{F}:\\R^n\\to\\R^m$ is continuously differentiable", "on an open set containing a compact set $D,$ then there is a constant", "$M$ such that", "\\begin{equation}\\label{eq:6.2.18}", "|\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X})|\\le M|\\mathbf{Y}-\\mathbf{X}|", "\\mbox{\\quad if\\quad}\\mathbf{X},\\mathbf{Y}\\in D.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "On", "$$", "S=\\set{(\\mathbf{X},\\mathbf{Y})}{\\mathbf{X},\\mathbf{Y}\\in D}\\subset \\R^{2n}", "$$", "define", "$$", "g(\\mathbf{X},\\mathbf{Y})=\\left\\{\\casespace\\begin{array}{ll}", "\\dst{\\frac{|\\mathbf{F}(\\mathbf{Y})-", "\\mathbf{F}(\\mathbf{X})", "-\\mathbf{F}'(\\mathbf{X})(\\mathbf{Y}-\\mathbf{X})|}{ |\\mathbf{Y}-\\mathbf{X}|}},&", "\\mathbf{Y}\\ne\\mathbf{X},\\\\[2\\jot]", " 0,&\\mathbf{Y}=\\mathbf{X}.\\end{array}\\right.", "$$", "Then $g$ is continuous for all $(\\mathbf{X},\\mathbf{Y})$ in $S$", "such that $\\mathbf{X}\\ne \\mathbf{Y}$. We now show that if $\\mathbf{X}_0\\in D$,", "then", "\\begin{equation}\\label{eq:6.2.19}", "\\lim_{(\\mathbf{X},\\mathbf{Y})\\to (\\mathbf{X}_0,\\mathbf{X}_0)}", "g(\\mathbf{X},\\mathbf{Y})=0", "=g(\\mathbf{X}_0,\\mathbf{X}_0);", "\\end{equation}", "that is, $g$ is also continuous at points $(\\mathbf{X}_0,\\mathbf{X}_0)$ in", "$S$.", "Suppose that $\\epsilon>0$ and $\\mathbf{X}_0\\in D$. Since the partial", "derivatives of $f_1$, $f_2$, \\dots, $f_m$ are continuous on an open", "set containing $D$, there is a $\\delta>0$ such that", "\\begin{equation}\\label{eq:6.2.20}", "\\left|\\frac{\\partial f_i(\\mathbf{Y})}{\\partial x_j}-\\frac{\\partial", "f_i(\\mathbf{X})", "}{\\partial x_j}\\right|<\\frac{\\epsilon}{\\sqrt{mn}}\\mbox{\\quad if\\quad}", "\\mathbf{X},\\mathbf{Y}\\in B_\\delta (\\mathbf{X}_0),\\ 1\\le i\\le m,\\", "1\\le j\\le n.", "\\end{equation}", "(Note that $\\partial f_i/\\partial x_j$ is uniformly continuous on", "$\\overline{B_\\delta(\\mathbf{X}_0)}$ for $\\delta$ sufficiently small, from", "Theorem~\\ref{thmtype:5.2.14}.) Applying", "Theorem~\\ref{thmtype:5.4.5}", "to $f_1$, $f_2$, \\dots, $f_m$, we find that if $\\mathbf{X}$, $\\mathbf{Y}\\in", "B_\\delta", "(\\mathbf{X}_0)$, then", "$$", "f_i(\\mathbf{Y})-f_i(\\mathbf{X})=\\sum_{j=1}^n", "\\frac{\\partial f_i(\\mathbf{X}_{i})}", "{\\partial x_j} (y_j-x_j),", "$$", "where $\\mathbf{X}_i$ is on the line segment from $\\mathbf{X}$ to $\\mathbf{Y}$.", "From this,", "\\begin{eqnarray*}", "\\left[f_i(\\mathbf{Y})-f_i(\\mathbf{X})", "-\\dst{\\sum_{j=1}^n}", "\\frac{\\partial f_i(\\mathbf{X})}{\\partial x_j} (y_j-x_j)\\right]^2", "\\ar=\\left[\\sum_{j=1}^n\\left[\\frac{\\partial f_i(\\mathbf{X}_i)}{\\partial", "x_j}-", "\\frac{\\partial f_i(\\mathbf{X})}{\\partial x_j}\\right] (y_j-x_j)\\right]^2\\\\", "\\ar\\le |\\mathbf{Y}-\\mathbf{X}|^2\\sum_{j=1}^n", "\\left[\\frac{\\partial f_i(\\mathbf{X}_{i})}", "{\\partial x_j}", "-\\frac{\\partial f_i(\\mathbf{X})}{\\partial x_j}\\right]^2\\\\", "\\ar{}\\mbox{(by Schwarz's inequality)}\\\\", "\\ar< \\frac{\\epsilon^2}{ m} |\\mathbf{Y}-\\mathbf{X}|^2\\quad\\mbox{\\quad (by", "\\eqref{eq:6.2.20})\\quad}.", "\\end{eqnarray*}", "Summing from $i=1$ to $i=m$ and taking square roots yields", "$$", "|\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X})-\\mathbf{F}'(\\mathbf{X})", "(\\mathbf{Y}-\\mathbf{X})|", "<\\epsilon |\\mathbf{Y}-\\mathbf{X}|\\mbox{\\quad if\\quad}", "\\mathbf{X},\\mathbf{Y}\\in B_\\delta(\\mathbf{X}_0).", "$$", "\\nopagebreak", "This implies \\eqref{eq:6.2.19} and completes the proof that $g$ is", "continuous on $S$.", "\\newpage", " Since $D$ is compact, so is $S$", "(Exercise~\\ref{exer:5.1.27}).", "Therefore, $g$ is bounded on $S$", "(Theorem~\\ref{thmtype:5.2.12}); thus, for some $M_1$,", "$$", "|\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X})-\\mathbf{F}'(\\mathbf{X}) (\\mathbf{Y}", "-\\mathbf{X})|\\le M_1|\\mathbf{X}-\\mathbf{Y}|\\mbox{\\quad if\\quad}", "\\mathbf{X},\\mathbf{Y}\\in D.", "$$", "But", "\\begin{equation}\\label{eq:6.2.21}", "\\begin{array}{rcl}", "|\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X}) |\\ar\\le", "|\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X})-\\mathbf{F}'(\\mathbf{X})", "(\\mathbf{Y}-\\mathbf{X})|+|\\mathbf{F}'(\\mathbf{X})(\\mathbf{Y}-\\mathbf{X})|\\\\", "\\ar\\le (M_1+\\|\\mathbf{F}'(\\mathbf{X})\\|) |(\\mathbf{Y}-\\mathbf{X}|.", "\\end{array}", "\\end{equation}", "Since", "$$", "\\|\\mathbf{F}'(\\mathbf{X})\\|", "\\le\\left(\\sum_{i=1}^m\\sum_{j=1}^n\\left[\\frac{\\partial", "f_i(\\mathbf{X}) }{\\partial x_j}\\right]^2\\right)^{1/2}", "$$", "and the partial derivatives $\\{\\partial f_i/\\partial x_j\\}$ are", "bounded on $D$, it follows that $\\|\\mathbf{F}'(\\mathbf{X})\\|$ is bounded on", "$D$; that is, there is a constant $M_2$ such that", "$$", "\\|\\mathbf{F}'(\\mathbf{X})\\|\\le M_2,\\quad\\mathbf{X}\\in D.", "$$", "Now \\eqref{eq:6.2.21} implies \\eqref{eq:6.2.18} with $M=M_1+M_2$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.2.14", "TRENCH_REAL_ANALYSIS-thmtype:5.4.5", "TRENCH_REAL_ANALYSIS-thmtype:5.2.12" ], "ref_ids": [ 154, 164, 152 ] } ], "ref_ids": [] }, { "id": 259, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.6", "categories": [], "title": "", "contents": [ "Suppose that $|f(\\mathbf{X})|\\le", "M$ if $\\mathbf{X}$ is in the rectangle", "$$", "R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n].", "$$", "Let ${\\bf P}=P_1\\times P_2\\times\\cdots\\times P_n$ and ${\\bf P}'=", "P_1'\\times P_2'\\times\\cdots\\times P_n'$ be partitions of $R,$ where", "$P_j'$ is obtained by adding $r_j$ partition points to $P_j,$", "$1\\le j\\le n.$ Then", "\\begin{equation}\\label{eq:7.1.16}", "S({\\bf P})\\ge S({\\bf P}')\\ge S({\\bf P})-2MV(R)\\left(\\sum_{j=1}^n", "\\frac{r_j}{ b_j-a_j}\\right)\\|{\\bf P}\\|", "\\end{equation}", "and", "\\begin{equation}\\label{eq:7.1.17}", "s({\\bf P})\\le s({\\bf P}')\\le s({\\bf P})+2MV(R)\\left(\\sum_{j=1}^n", "\\frac{r_j", "}{ b_j-a_j}\\right)\\|{\\bf P}\\|.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "We will prove", " \\eqref{eq:7.1.16} and leave the proof of \\eqref{eq:7.1.17} to you", "(Exercise~\\ref{exer:7.1.7}).", "First suppose that", " $P_1'$ is obtained by adding one point to $P_1$, and", "$P_j'=P_j$ for $2\\le j\\le n$.", "If $P_r$ is", "defined by", "$$", "P_r: a_r=a_{r0}0,$ there is", " a $\\delta>0$ such that", "\\vspace{4pt}", "$$", "\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\le S({\\bf P})<\\overline{\\int_R}\\,", "f(\\mathbf{X})\\,d\\mathbf{X}+\\epsilon", "$$", "\\vspace{4pt}", "and", "\\vspace{4pt}", "$$", "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\ge s({\\bf P})>", "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}-\\epsilon", "$$", "\\vspace{4pt}", "if $\\|{\\bf P}\\|<\\delta.$" ], "refs": [], "proofs": [ { "contents": [ "Exercise~\\ref{exer:7.1.10}.", "The next theorem is analogous to Theorem~3.2.5.", "\\begin{theorem}\\label{thmtype:7.1.10}", "If $f$ is bounded on a rectangle $R$ and", "\\vspace{2pt}", "$$", "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=", "\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=L,", "$$", "\\vspace{2pt}", "then $f$ is integrable on $R,$ and", "\\vspace{2pt}", "$$", "\\int_R f(\\mathbf{X})\\,d\\mathbf{X}=L.", "$$", "\\end{theorem}" ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 261, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.15", "categories": [], "title": "", "contents": [ "The union of finitely many sets with zero content has zero content$.$" ], "refs": [], "proofs": [ { "contents": [ "Suppose that $\\epsilon>0$. Since $E$ has zero content, there are", "rectangles", "$T_1$, $T_2$, \\dots, $T_m$ such that", "\\begin{equation} \\label{eq:7.1.31}", "E\\subset\\bigcup_{j=1}^m T_j", "\\end{equation}", "and", "\\begin{equation} \\label{eq:7.1.32}", "\\sum_{j=1}^m V(T_j)<\\epsilon.", "\\end{equation}", " We may assume that", "$T_1$, $T_2$, \\dots, $T_m$ are contained in $R$, since, if not, their", "intersections with", "$R$ would be contained in $R$, and still satisfy \\eqref{eq:7.1.31}", "and \\eqref{eq:7.1.32}.", " We may also assume that if $T$ is any rectangle such", "that", "\\begin{equation}\\label{eq:7.1.33}", "T\\bigcap\\left(\\bigcup_{j=1}^m T_j^0\\right)=\\emptyset, \\mbox{\\quad", "then", "\\quad}T\\cap E=\\emptyset", "\\end{equation}", "\\newpage", "\\noindent", "since if this were not so, we could make it so by enlarging", "$T_1$, $T_2$, \\dots, $T_m$", "slightly while maintaining \\eqref{eq:7.1.32}. Now suppose that", "\\vspace*{1pt}", "$$", "T_j=[a_{1j},b_{1j}]\\times [a_{2j},b_{2j}]\\times\\cdots\\times", "[a_{nj},b_{nj}],\\quad 1\\le j\\le m,", "$$", "\\vspace*{1pt}", "\\noindent let $P_{i0}$ be the partition of $[a_i,b_i]$ (see", "\\eqref{eq:7.1.30}) with partition points", "$$", "a_i,b_i,a_{i1},b_{i1},a_{i2},b_{i2}, \\dots,a_{im},b_{im}", "\\vspace*{1pt}", "$$", "(these are not in increasing order), $1\\le i\\le n$, and let", "\\vspace*{1pt}", "$$", "{\\bf P}_0=P_{10}\\times P_{20}\\times\\cdots\\times P_{n0}.", "$$", "\\vspace*{1pt}", "\\noindent\\hskip-.3em Then ${\\bf P}_0$ consists of rectangles whose", "union equals $\\cup_{j=1}^m T_j$", "and other rectangles", "$T'_1$, $T'_2$, \\dots, $T'_k$ that do not intersect $E$. (We need", "\\eqref{eq:7.1.33} to be sure that $T'_i\\cap E=\\emptyset,", "1\\le i\\le k.)$ If we let", "$$", "B=\\bigcup_{j=1}^m T_j\\mbox{\\quad and\\quad} C=\\bigcup^k_{i=1} T'_i,", "$$", "then $R=B\\cup C$ and $f$ is continuous on the compact set $C$.", "If ${\\bf P}=\\{R_1,R_2, \\dots,R_k\\}$ is a refinement of ${\\bf P}_0$,", "then every subrectangle $R_j$ of ${\\bf P}$ is contained entirely in", "$B$ or entirely in $C$. Therefore, we can write", "\\vspace*{1pt}", "\\begin{equation}\\label{eq:7.1.34}", "S({\\bf P})-s({\\bf P})=\\Sigma_1(M_j-m_j)", "V(R_j)+\\Sigma_2(M_j-m_j)V(R_j),", "\\end{equation}", "\\vspace*{1pt}", "\\noindent \\hskip-.3em", "where $\\Sigma_1$ and $\\Sigma_2$ are summations over values of $j$ for", "which $R_j\\subset B$ and $R_j\\subset C$, respectively. Now suppose that", "$$", "|f(\\mathbf{X})|\\le M\\mbox{\\quad for $\\mathbf{X}$ in $R$}.", "$$", "Then", "\\begin{equation}\\label{eq:7.1.35}", "\\Sigma_1(M_j-m_j) V(R_j)\\le2M\\,\\Sigma_1 V(R_j)=2M\\sum_{j=1}^m V(T_j)<", "2M\\epsilon,", "\\end{equation}", "from \\eqref{eq:7.1.32}.", "Since $f$ is uniformly continuous on the compact set $C$", "(Theorem~\\ref{thmtype:5.2.14}),", "there is a $\\delta>0$ such that $M_j-m_j<\\epsilon$ if", "$\\|{\\bf P}\\|< \\delta$ and $R_j\\subset C$; hence,", "$$", "\\Sigma_2(M_j-m_j)V(R_j)<\\epsilon\\Sigma_2\\, V(R_j)\\le\\epsilon V(R).", "$$", "This, \\eqref{eq:7.1.34}, and \\eqref{eq:7.1.35} imply that", "$$", "S({\\bf P})-s({\\bf P})<[2M+V(R)]\\epsilon", "$$", "if $\\|{\\bf P}\\|<\\delta$ and ${\\bf P}$ is a refinement of ${\\bf P}_0$.", "Therefore, Theorem~\\ref{thmtype:7.1.12} implies that $f$ is integrable on", "$R$.", "\\enlargethispage{4\\baselineskip}" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.2.14", "TRENCH_REAL_ANALYSIS-thmtype:7.1.12" ], "ref_ids": [ 154, 195 ] } ], "ref_ids": [] }, { "id": 262, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.29", "categories": [], "title": "", "contents": [ "Suppose that $S$ is contained in a bounded set $T$ and $f$ is integrable", "on $S.$ Then", " $f_S$ $($see $\\eqref{eq:7.1.36})$ is integrable on $T,$ and", "$$", "\\int_T f_S(\\mathbf{X})\\,d\\mathbf{X}=\\int_S f(\\mathbf{X})\\,d\\mathbf{X}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "From Definition~\\ref{thmtype:7.1.17} with $f$ and $S$ replaced by $f_S$", "and $T$,", "\\pagebreak", "$$", "(f_S)_T(\\mathbf{X})=\\left\\{\\casespace\\begin{array}{ll} f_S(\\mathbf{X}),&\\mathbf{X}\\in T,\\\\", " 0,&\\mathbf{X}\\not\\in T.\\end{array}\\right.", "$$", " Since $S\\subset T$, $(f_S)_T=f_S$.", "(Verify.) Now suppose that $R$ is a rectangle containing $T$.", " Then $R$ also", "contains $S$ (Figure~\\ref{figure:7.1.7}),", " \\vspace*{12pt}", " \\centereps{2.3in}{1.45in}{fig070107.eps}", " \\vskip6pt", " \\refstepcounter{figure}", " \\centerline{\\bf Figure \\thefigure} \\label{figure:7.1.7}", " \\vskip12pt", "\\noindent so", "$$", "\\begin{array}{rcll}", "\\dst\\int_Sf(\\mathbf{X})\\,d\\mathbf{X}\\ar=\\dst\\int_Rf_S(\\mathbf{X})\\,d\\mathbf{X}&", "\\mbox{(Definition~\\ref{thmtype:7.1.17}, applied to $f$ and $S$})\\\\[4\\jot]", "\\ar=\\dst\\int_R(f_S)_T(\\mathbf{X})\\,d\\mathbf{X}&", "\\mbox{(since $(f_S)_T=f_S$)}\\\\[4\\jot]", "\\ar=\\dst\\int_Tf_S(\\mathbf{X})\\,d\\mathbf{X}&", "\\mbox{(Definition~\\ref{thmtype:7.1.17}, applied to $f_S$ and $T$}),", "\\end{array}", "$$", "which completes the proof." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.1.17", "TRENCH_REAL_ANALYSIS-thmtype:7.1.17", "TRENCH_REAL_ANALYSIS-thmtype:7.1.17" ], "ref_ids": [ 362, 362, 362 ] } ], "ref_ids": [] }, { "id": 263, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.3", "categories": [], "title": "", "contents": [ "Suppose that $K$ is a bounded set with zero content and $\\epsilon,$", "$\\rho>0.$ Then there are cubes $C_1,$ $C_2,$ \\dots$,$", "$C_r$ with edge lengths", "$<\\rho$ such that $C_j\\cap K\\ne\\emptyset,$ $1\\le j\\le r,$", "\\begin{equation}\\label{eq:7.3.5}", "K\\subset\\bigcup_{j=1}^r C_j,", "\\end{equation}", "and", "$$", "\\sum_{j=1}^r V(C_j)<\\epsilon.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $V(K)=0$,", "$$", "\\int_C\\psi_K(\\mathbf{X})\\,d\\mathbf{X}=0", "$$", "if $C$ is any cube containing $K$. From this and the", "definition of the integral, there is a $\\delta>0$ such that if ${\\bf", "P}$ is any partition of $C$ with $\\|{\\bf P}\\|\\le\\delta$ and $\\sigma$", "is any Riemann sum of $\\psi_K$ over ${\\bf P}$, then", "\\begin{equation}\\label{eq:7.3.6}", "0\\le\\sigma\\le\\epsilon.", "\\end{equation}", "\\newpage", "\\noindent", "Now suppose that ${\\bf P}=\\{C_1,C_2,\\dots,C_k\\}$ is a partition of $C$", "into cubes with", "\\begin{equation}\\label{eq:7.3.7}", "\\|{\\bf P}\\|<\\min (\\rho,\\delta),", "\\end{equation}", "and let $C_1$, $C_2$, \\dots, $C_k$ be numbered so that $C_j\\cap K\\ne", "\\emptyset$ if $1\\le j\\le r$ and", "$C_j\\cap K=\\emptyset$ if $r+1\\le j\\le k$. Then \\eqref{eq:7.3.5} holds, and", "a typical Riemann sum of $\\psi_K$ over ${\\bf P}$ is of the form", "$$", "\\sigma=\\sum_{j=1}^r\\psi_K(\\mathbf{X}_j)V(C_j)", "$$", "with $\\mathbf{X}_j\\in C_j$, $1\\le j\\le r$. In particular, we", "can choose", "$\\mathbf{X}_j$ from $K$, so that $\\psi_K(\\mathbf{X}_j)=1$, and", "$$", "\\sigma=\\sum_{j=1}^r V(C_j).", "$$", "Now \\eqref{eq:7.3.6} and \\eqref{eq:7.3.7} imply that $C_1$, $C_2$, \\dots,", "$C_r$ have the required properties." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 264, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.4", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{G}: \\R^n\\to \\R^n$ is continuously", "differentiable on a bounded open set $S,$ and let $K$ be a closed", "subset of $S$ with zero content$.$ Then $\\mathbf{G}(K)$ has zero content." ], "refs": [], "proofs": [ { "contents": [ "Since $K$ is a compact subset of the open set $S$, there is a", " $\\rho_1>0$ such that the compact set", "$$", "K_{\\rho_1}=\\set{\\mathbf{X}}{\\dist(\\mathbf{X},K)\\le\\rho_1}", "$$", "is contained in $S$ (Exercise~5.1.26).", "From", "Lemma~\\ref{thmtype:6.2.7}, there is a constant $M$ such that", "\\begin{equation}\\label{eq:7.3.8}", "|\\mathbf{G}(\\mathbf{Y})-\\mathbf{G}(\\mathbf{X})|\\le M|\\mathbf{Y}-\\mathbf{X}|", "\\mbox{\\quad if\\quad}\\mathbf{X},\\mathbf{Y}\\in K_{\\rho_1}.", "\\end{equation}", "Now suppose that $\\epsilon>0$. Since $V(K)=0$,", "there are cubes $C_1$, $C_2$, \\dots, $C_r$ with edge", "lengths", "$s_1$, $s_2$, \\dots, $s_r<\\rho_1/\\sqrt n$ such that $C_j\\cap", "K\\ne\\emptyset$, $1\\le j\\le r$,", "$$", "K\\subset\\bigcup_{j=1}^r C_j,", "$$", "and", "\\begin{equation} \\label{eq:7.3.9}", "\\sum_{j=1}^r V(C_j)<\\epsilon", "\\end{equation}", "(Lemma~\\ref{thmtype:7.3.3}). For $1\\le j\\le r$, let $\\mathbf{X}_j\\in C_j\\cap", "K$. If $\\mathbf{X}\\in C_j$, then", "$$", "|\\mathbf{X}-\\mathbf{X}_j|\\le s_j\\sqrt n<\\rho_1,", "$$", "\\newpage", "\\noindent", "so $\\mathbf{X}\\in K$ and", "$|\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{X}_j)|\\le M|\\mathbf{X}-\\mathbf{X}_j|\\le", "M\\sqrt{n}\\,s_j$,", "from \\eqref{eq:7.3.8}.", "Therefore, $\\mathbf{G}(C_j)$ is contained in a cube", "$\\widetilde{C}_j$ with edge length $2M\\sqrt{n}\\,s_j$,", " centered at $\\mathbf{G}(\\mathbf{X}_j)$. Since", "$$", "V(\\widetilde{C}_j)=(2M\\sqrt{n})^ns_j^n=(2M\\sqrt{n})^nV(C_j),", "$$", "we now see that", "$$", "\\mathbf{G}(K)\\subset\\bigcup_{j=1}^r\\widetilde{C}_j", "$$", "and", "$$", "\\sum_{j=1}^r V(\\widetilde{C}_j)\\le", "(2M\\sqrt{n})^n\\sum_{j=1}^r V(C_j)<(2M\\sqrt{n})^n\\epsilon,", "$$", "where the last inequality follows from \\eqref{eq:7.3.9}.", "Since $(2M\\sqrt{n})^n$ does not depend on $\\epsilon$, it follows", "that $V(\\mathbf{G}(K))=0$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:6.2.7", "TRENCH_REAL_ANALYSIS-thmtype:7.3.3" ], "ref_ids": [ 258, 263 ] } ], "ref_ids": [] }, { "id": 265, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.6", "categories": [], "title": "", "contents": [ "A nonsingular $n\\times n$ matrix", "$\\mathbf{A}$ can be written as", "\\begin{equation}\\label{eq:7.3.10}", "\\mathbf{A}=\\mathbf{E}_k\\mathbf{E}_{k-1}\\cdots\\mathbf{E}_1,", "\\end{equation}", "where each $\\mathbf{E}_i$ is a matrix that can be obtained from the", "$n\\times n$ identity matrix $\\mathbf{I}$ by one of the following", "operations$:$", "\\begin{alist}", "\\item % (a)", "interchanging two rows of $\\mathbf{I};$", "\\item % (b)", "multiplying a row of $\\mathbf{I}$ by a nonzero constant$;$", "\\item % (c)", "adding a multiple of one row of $\\mathbf{I}$ to another$.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "Theorem~\\ref{thmtype:7.3.5} implies that $\\mathbf{L}(S)$ is", "Jordan measurable. If", "\\begin{equation} \\label{eq:7.3.15}", "V(\\mathbf{L}(R))=|\\det(\\mathbf{A})| V(R)", "\\end{equation}", "whenever $R$ is a rectangle, then", " \\eqref{eq:7.3.14} holds if $S$", "is any compact Jordan measurable set. To see this, suppose that", "$\\epsilon>0$, let", "$R$ be a rectangle containing $S$, and let", "${\\bf P}=\\{R_1,R_2,\\dots,R_k\\}$ be a partition of $R$ such that the", "upper and lower sums of $\\psi_S$ over ${\\bf", "P}$ satisfy the inequality", "\\begin{equation}\\label{eq:7.3.16}", "S({\\bf P})-s({\\bf P})<\\epsilon.", "\\end{equation}", "Let ${\\mathcal U}_1$ and ${\\mathcal U}_2$ be as in \\eqref{eq:7.3.2}.", "From \\eqref{eq:7.3.3} and \\eqref{eq:7.3.4},", "\\begin{equation}\\label{eq:7.3.17}", "s({\\bf P})=\\sum_{j\\in{\\mathcal U}_1} V(R_j)\\le V(S)\\le\\sum_{j\\in{\\mathcal U}_1} V(R_j)+\\sum_{j\\in{\\mathcal U}_2}", "V(R_j)=S({\\bf P}).", "\\end{equation}", " Theorem~\\ref{thmtype:7.3.7}", "implies that $\\mathbf{L}(R_1)$, $\\mathbf{L}(R_2)$, \\dots, $\\mathbf{L}(R_k)$", "and", "$\\mathbf{L}(S)$ are all Jordan measurable.", "Since", "$$", "\\bigcup_{j\\in{\\mathcal U}_1}R_j\\subset S\\subset\\bigcup_{j\\in{\\mathcal", "S}_1\\cup{\\mathcal S_2}}R_j,", "$$", "it follows that", "$$", "L\\left(\\bigcup_{j\\in{\\mathcal U}_1}R_j\\right)\\subset", "L(S)\\subset L\\left(\\bigcup_{j\\in{\\mathcal S}_1\\cup{\\mathcal S_2}}R_j\\right).", "$$", "Since $L$ is one-to-one on $\\R^n$, this implies that", "\\begin{equation} \\label{eq:7.3.18}", "\\sum_{j\\in{\\mathcal U}_1} V(\\mathbf{L}(R_j))\\le V(\\mathbf{L}(S))\\le\\sum_{j\\in{\\mathcal U}_1}", "V(\\mathbf{L}(R_j))+\\sum_{j\\in{\\mathcal U}_2} V(\\mathbf{L}(R_j)).", "\\end{equation}", "If we assume that \\eqref{eq:7.3.15} holds whenever $R$ is a rectangle,", "then", "$$", "V(\\mathbf{L}(R_j))=|\\det(\\mathbf{A})|V(R_j),\\quad 1\\le j\\le k,", "$$", "so \\eqref{eq:7.3.18} implies that", "$$", "s({\\bf P})\\le \\frac{V(\\mathbf{L}(S))}{ |\\det(\\mathbf{A})|}\\le S({\\bf P}).", "$$", "This, \\eqref{eq:7.3.16} and \\eqref{eq:7.3.17} imply that", "$$", "\\left|V(S)-\\frac{V(\\mathbf{L}(S))}{ |\\det(\\mathbf{A})|}\\right|<\\epsilon;", "$$", "hence, since $\\epsilon$ can be made arbitrarily small, \\eqref{eq:7.3.14}", "follows for any Jordan measurable set.", "To complete the proof, we must verify \\eqref{eq:7.3.15} for every", "rectangle", "$$", "R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n]=I_1\\times", "I_2\\times\\cdots\\times I_n.", "$$", " Suppose that $\\mathbf{A}$ in \\eqref{eq:7.3.12} is an elementary matrix;", "that is, let", "$$", "\\mathbf{X}=\\mathbf{L}(\\mathbf{Y})=\\mathbf{EY}.", "$$", "{\\sc Case 1}. If $\\mathbf{E}$ is obtained by interchanging the $i$th and", "$j$th rows of $\\mathbf{I}$, then", "$$", "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$ and $r\\ne j$};\\\\", "y_j&\\mbox{if $r=i$};\\\\", "y_i&\\mbox{if $r=j$}.\\end{array}\\right.", "$$", "Then $\\mathbf{L}(R)$ is the Cartesian product of $I_1$,", "$I_2$, \\dots, $I_n$ with", "$I_i$ and $I_j$ interchanged, so", "$$", "V(\\mathbf{L}(R))=V(R)=|\\det(\\mathbf{E})|V(R)", "$$", "since $\\det(\\mathbf{E})=-1$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", "{\\sc Case 2}. If $\\mathbf{E}$ is obtained by multiplying the $r$th row of", "$\\mathbf{I}$ by $a$, then", "$$", "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$},\\\\", "ay_i&\\mbox{if $r=i$}.\\end{array}\\right.", "$$", "Then", "$$", "\\mathbf{L}(R)=I_1\\times\\cdots\\times I_{i-1}\\times I'_i\\times I_{i+1}\\times", "\\cdots\\times I_n,", "$$", "where $I'_i$ is an interval with length equal to $|a|$ times the", "length of $I_i$, so", "$$", "V(\\mathbf{L}(R))=|a|V(R)=|\\det(\\mathbf{E})|V(R)", "$$", "since $\\det(\\mathbf{E})=a$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", "{\\sc Case 3}. If $\\mathbf{E}$ is obtained by adding $a$ times the $j$th", "row of $\\mathbf{I}$ to its $i$th row ($j\\ne i$), then", "$$", "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$};\\\\", "y_i+ay_j&\\mbox{if $r=i$}.\\end{array}\\right.", "$$", "Then", "$$", "\\mathbf{L}(R)=\\set{(x_1,x_2,\\dots,x_n)}{a_i+ax_j\\le x_i\\le b_i+ax_j", "\\mbox{ and } a_r\\le x_r\\le b_r\\mbox{if } r\\ne i},", "$$", "which is a parallelogram if $n=2$ and a parallelepiped if $n=3$", "(Figure~\\ref{figure:7.3.1}). Now", "$$", "V(\\mathbf{L}(R))=\\int_{\\mathbf{L}(R)} d\\mathbf{X},", "$$", "which we can evaluate as an iterated integral in which the first", "integration is with respect to $x_i$. For example, if $i=1$, then", "\\begin{equation}\\label{eq:7.3.19}", "V(\\mathbf{L}(R))=\\int^{b_n}_{a_n} dx_n\\int^{b_{n-1}}_{a_{n-1}}", "dx_{n-1}\\cdots\\int^{b_2}_{a_2} dx_2\\int^{b_1+ax_j}_{a_1+ax_j} dx_1.", "\\end{equation}", "\\newpage", "\\noindent", "Since", "$$", "\\int^{b_1+ax_j}_{a_1+ax_j} dy_1=\\int^{b_1}_{a_1} dy_1,", "$$", "\\eqref{eq:7.3.19} can be rewritten as", "\\begin{eqnarray*}", "V(\\mathbf{L}(R))\\ar=\\int^{b_n}_{a_n} dx_n\\int^{b_{n-1}}_{a_{n-1}}", "dx_{n-1}\\cdots\\int^{b_2}_{a_2} dx_2\\int^{b_1}_{a_1} dx_1\\\\", "\\ar=(b_n-a_n)(b_{n-1}-a_{n-1})\\cdots (b_1-a_1)=V(R).", "\\end{eqnarray*}", " Hence,", "$V(\\mathbf{L}(R))=|\\det(\\mathbf{E})|V(R)$,", "since $\\det(\\mathbf{E})=1$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", "\\vskip12pt", " \\centereps{3.6in}{4.6in}{fig070301.eps}", " \\vskip6pt", " \\refstepcounter{figure}", " \\centerline{\\bf Figure \\thefigure} \\label{figure:7.3.1}", " \\vskip12pt", "From what we have shown so far, \\eqref{eq:7.3.14} holds if $\\mathbf{A}$ is an", "elementary matrix and $S$ is any compact Jordan measurable set. If", "$\\mathbf{A}$ is an arbitrary nonsingular matrix,", "\\newpage", "\\noindent", "\\hskip -.0em", "then we can write $\\mathbf{A}$", "as a product of elementary matrices \\eqref{eq:7.3.10} and apply our known", "result successively to $\\mathbf{L}_1$, $\\mathbf{L}_2$, \\dots, $\\mathbf{L}_k$", "(see", "\\eqref{eq:7.3.13}). This yields", "$$", "V(\\mathbf{L}(S))=|\\det(\\mathbf{E}_k)|\\,|\\det(\\mathbf{E}_{k-1})|\\cdots", "|\\det\\mathbf{E}_1| V(S)=|\\det(\\mathbf{A})|V(S),", "$$", "by Theorem~\\ref{thmtype:6.1.9} and induction." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.3.5", "TRENCH_REAL_ANALYSIS-thmtype:7.3.7", "TRENCH_REAL_ANALYSIS-thmtype:6.1.9" ], "ref_ids": [ 215, 216, 173 ] } ], "ref_ids": [] }, { "id": 266, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.10", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{G}:\\E^n\\to \\R^n$ is regular", " on a cube $C$ in $\\E^n,$ and let $\\mathbf{A}$ be a", "nonsingular $n\\times n$ matrix$.$ Then", "\\begin{equation}\\label{eq:7.3.29}", "V(\\mathbf{G}(C))\\le |\\det(\\mathbf{A})|\\left[\\max", "\\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C}", "\\right]^n V(C).", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Let $s$ be the edge length of $C$. Let $\\mathbf{Y}_0=", "(c_1,c_2,\\dots,c_n)$ be the center of $C$, and suppose that", " $\\mathbf{H}=(y_1,y_2,\\dots,y_n)\\in C$.", "If $\\mathbf{H}= (h_1,h_2,\\dots,h_n)$ is continuously differentiable on", "$C$, then applying the mean value theorem", "(Theorem~\\ref{thmtype:5.4.5}) to the components of", "$\\mathbf{H}$ yields", "$$", "h_i(\\mathbf{Y})-h_i(\\mathbf{Y}_0)=\\sum_{j=1}^n", "\\frac{\\partial h_i(\\mathbf{Y}_i)}{\\partial y_j}(y_j-c_j),\\quad 1\\le i\\le n,", "$$", "where $\\mathbf{Y}_i\\in C$. Hence, recalling that", "$$", "\\mathbf{H}'(\\mathbf{Y})=\\left[\\frac{\\partial h_i}{\\partial", "y_j}\\right]_{i,j=1}^n,", "$$", "applying Definition~\\ref{thmtype:7.3.9}, and noting that $|y_j-c_j|\\le", "s/2$, $1\\le j\\le n$, we infer that", "$$", "|h_i(\\mathbf{Y})-h_i(\\mathbf{Y}_0)|\\le \\frac{s}{2}", "\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C},\\quad 1\\le i\\le", "n.", "$$", "This means that $\\mathbf{H}(C)$ is", "contained in a cube with center $\\mathbf{X}_0=\\mathbf{H}(\\mathbf{Y}_0)$ and edge", " length", "$$", "s\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C}.", "$$", "Therefore,", "\\begin{equation}\\label{eq:7.3.30}", "\\begin{array}{rcl}", "V(\\mathbf{H}(C))\\ar\\le", "\\left[\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty\\right]^n}{\\mathbf{Y}\\in", "C} s^n\\\\[2\\jot]", "\\ar=\\left[\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty\\right]^n}{\\mathbf{Y}\\in C}", "V(C).", "\\end{array}", "\\end{equation}", "Now let", "$$", "\\mathbf{L}(\\mathbf{X})=\\mathbf{A}^{-1}\\mathbf{X}", "$$", "and set $\\mathbf{H}=\\mathbf{L}\\circ\\mathbf{G}$; then", "$$", "\\mathbf{H}(C)=\\mathbf{L}(\\mathbf{G}(C))", "\\mbox{\\quad and\\quad}\\mathbf{H}'=\\mathbf{A}^{-1}\\mathbf{G}',", "$$", "so \\eqref{eq:7.3.30} implies that", "\\begin{equation}\\label{eq:7.3.31}", "V(\\mathbf{L}(\\mathbf{G}(C)))\\le", "\\left[\\max\\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C}", "\\right]^nV(C).", "\\end{equation}", "Since $\\mathbf{L}$ is linear,", "Theorem~\\ref{thmtype:7.3.7} with $\\mathbf{A}$ replaced by $\\mathbf{A}^{-1}$ implies that", "$$", "V(\\mathbf{L}(\\mathbf{G}(C)))=|\\det(\\mathbf{A})^{-1}|V(\\mathbf{G}(C)).", "$$", "This and \\eqref{eq:7.3.31} imply that", "$$", "|\\det(\\mathbf{A}^{-1})|V(\\mathbf{G}(C))", "\\le\\left[\\max\\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in", "C}", "\\right]^nV(C).", "$$", "Since $\\det(\\mathbf{A}^{-1})=1/\\det(\\mathbf{A})$, this", "implies \\eqref{eq:7.3.29}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.4.5", "TRENCH_REAL_ANALYSIS-thmtype:7.3.9", "TRENCH_REAL_ANALYSIS-thmtype:7.3.7" ], "ref_ids": [ 164, 365, 216 ] } ], "ref_ids": [] }, { "id": 267, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.11", "categories": [], "title": "", "contents": [ "If $\\mathbf{G}:\\E^n\\rightarrow \\R^n$", " is regular on a cube $C$ in $\\R^n,$ then", "\\begin{equation}\\label{eq:7.3.32}", "V(\\mathbf{G}(C))\\le\\int_C |JG(\\mathbf{Y})|\\,d\\mathbf{Y}.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Let ${\\bf P}$ be a partition of $C$ into subcubes $C_1$, $C_2$,", "\\dots, $C_k$ with centers $\\mathbf{Y}_1$, $\\mathbf{Y}_2$,", "\\dots, $\\mathbf{Y}_k$. Then", "\\begin{equation}\\label{eq:7.3.33}", "V(\\mathbf{G}(C))=\\sum_{j=1}^k V(\\mathbf{G}(C_j)).", "\\end{equation}", "Applying Lemma~\\ref{thmtype:7.3.10}", "to $C_j$ with $\\mathbf{A}=\\mathbf{G}'(\\mathbf{A}_j)$ yields", "\\begin{equation}\\label{eq:7.3.34}", "V(\\mathbf{G}(C_j))\\le |J\\mathbf{G}(\\mathbf{Y}_j)|", "\\left[\\max\\set{\\|(\\mathbf{G}'(\\mathbf{Y}_j))^{-1}", "\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C_j}", "\\right]^n V(C_j).", "\\end{equation}", "Exercise~\\ref{exer:6.1.22} implies that if $\\epsilon>0$, there", "is a $\\delta>0$ such that", "$$", "\\max\\set{\\|(\\mathbf{G}'(\\mathbf{Y}_j))^{-1}", "\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C_j}", "<1+\\epsilon,\\quad 1\\le j\\le k,\\mbox{\\quad if\\quad}\\|{\\bf P}\\|<\\delta.", "$$", "Therefore, from \\eqref{eq:7.3.34},", "$$", "V(\\mathbf{G}(C_j))\\le (1+\\epsilon)^n|J\\mathbf{G}(\\mathbf{Y}_j)|V(C_j),", "$$", " so \\eqref{eq:7.3.33} implies that", "$$", "V(\\mathbf{G}(C))\\le (1+\\epsilon)^n\\sum_{j=1}^k", "|J\\mathbf{G}(\\mathbf{Y}_j)|V(C_j)\\mbox{\\quad if\\quad}\\|{\\bf P}\\|<\\delta.", "$$", "Since the sum on the right is a Riemann sum for", " $\\int_C |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}$ and $\\epsilon$ can be", "taken arbitrarily small, this implies \\eqref{eq:7.3.32}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.3.10" ], "ref_ids": [ 266 ] } ], "ref_ids": [] }, { "id": 268, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.12", "categories": [], "title": "", "contents": [ " Suppose that $S$ is Jordan measurable", "and $\\epsilon,$ $\\rho>0.$ Then there are cubes", "$C_1,$ $C_2,$ \\dots$,$ $C_r$ in $S$ with edge lengths $<\\rho,$ such", "that $C_j\\subset S,$ $1\\le j\\le r,$", "$C_i^0\\cap C_j^0=\\emptyset$ if $i\\ne j,$ and", "\\begin{equation} \\label{eq:7.3.35}", "V(S)\\le\\sum_{j=1}^r V(C_j)+\\epsilon.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Since $S$ is Jordan measurable,", "$$", "\\int_C\\psi_S(\\mathbf{X})\\,d\\mathbf{X}=V(S)", "$$", "if $C$ is any cube containing $S$. From this and the", "definition of the integral, there is a $\\delta>0$ such that if ${\\bf", "P}$ is any partition of $C$ with $\\|{\\bf P}\\|<\\delta$ and $\\sigma$", "is any Riemann sum of $\\psi_S$ over ${\\bf P}$, then", "$\\sigma>V(S)-\\epsilon/2$. Therefore, if $s(P)$ is the lower sum of", "$\\psi_S$ over $\\mathbf{P}$, then", "\\begin{equation} \\label{eq:7.3.36}", "s(\\mathbf{P})>V(S)-\\epsilon\\mbox{\\quad if \\quad}\\|\\mathbf{P}\\|<\\delta.", "\\end{equation}", "Now suppose that ${\\bf P}=\\{C_1,C_2,\\dots,C_k\\}$ is a partition of $C$", "into cubes with", "$\\|{\\bf P}\\|<\\min (\\rho,\\delta)$,", "and let $C_1$, $C_2$, \\dots, $C_k$ be numbered so that $C_j\\subset", "S$ if", " $1\\le j\\le r$ and $C_j\\cap S^c\\ne\\emptyset$ if $j>r$.", "From \\eqref{eq:7.3.4}, $s(\\mathbf{P})=\\sum_{j=1}^rV(C_k)$. This and", "\\eqref{eq:7.3.36} imply \\eqref{eq:7.3.35}. Clearly, $C_i^0\\cap", "C_j^0=\\emptyset$ if $i\\ne j$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 269, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.13", "categories": [], "title": "", "contents": [ "Suppose that $\\mathbf{G}: \\E^n\\to \\R^n$ is regular on a", "compact Jordan measurable set $S$ and $f$ is continuous and", "nonnegative on", "$\\mathbf{G}(S).$", "Let", "\\begin{equation}\\label{eq:7.3.37}", "Q(S)=\\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}-\\int_S", " f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}.", "\\end{equation}", "Then $Q(S)\\le0.$" ], "refs": [], "proofs": [ { "contents": [ "From the continuity of $J\\mathbf{G}$ and $f$ on the compact sets $S$ and", "$\\mathbf{G}(S)$, there are constants $M_1$ and $M_2$ such that", "\\begin{equation}\\label{eq:7.3.38}", "|J\\mathbf{G}(\\mathbf{Y})|\\le M_1\\mbox{\\quad if\\quad}\\mathbf{Y}\\in S", "\\end{equation}", "and", "\\begin{equation}\\label{eq:7.3.39}", "|f(\\mathbf{X})|\\le M_2\\mbox{\\quad if\\quad}\\mathbf{X}\\in\\mathbf{G}(S)", "\\end{equation}", " (Theorem~\\ref{thmtype:5.2.11}).", "Now suppose that $\\epsilon>0$. Since", "$f\\circ\\mathbf{G}$ is uniformly continuous on $S$", "(Theorem~\\ref{thmtype:5.2.14}),", " there is a $\\delta>0$ such that", "\\begin{equation} \\label{eq:7.3.40}", "|f(\\mathbf{G}(\\mathbf{Y}))-f(\\mathbf{G}(\\mathbf{Y}'))|<\\epsilon", "\\mbox{\\quad if \\quad$|\\mathbf{Y}-\\mathbf{Y}'|<\\delta$", "and }\\mathbf{Y},\\mathbf{Y}' \\in S.", "\\end{equation}", "Now let $C_1$, $C_2$, \\dots, $C_r$ be chosen as described in", "Lemma~\\ref{thmtype:7.3.12}, with $\\rho=\\delta/\\sqrt{n}$.", " Let", "$$", "S_1=\\set{\\mathbf{Y}\\in S}{\\mathbf{Y}\\notin\\bigcup_{j=1}^r C_j}.", "$$", "Then $V(S_1)<\\epsilon$ and", "\\begin{equation} \\label{eq:7.3.41}", "S=\\left(\\bigcup_{j=1}^r C_j\\right)\\cup S_1.", "\\end{equation}", "Suppose that $\\mathbf{Y}_1$, $\\mathbf{Y}_2$, \\dots, $\\mathbf{Y}_r$ are points in", "$C_1$, $C_2$, \\dots, $C_r$ and $\\mathbf{X}_j=\\mathbf{G}(\\mathbf{Y}_j)$, $1\\le", "j\\le r$. From", "\\eqref{eq:7.3.41} and Theorem~\\ref{thmtype:7.1.30},", "\\begin{eqnarray*}", "Q(S)\\ar=\\int_{\\mathbf{G}(S_1)} f(\\mathbf{X})\\,d\\mathbf{X}-\\int_{S_1}", "f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y} \\\\", "\\ar{}+\\sum_{j=1}^r\\int_{\\mathbf{G}(C_j)} f(\\mathbf{X})\\,d\\mathbf{X}-", "\\sum_{j=1}^r\\int_{C_j}", "f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\\\", "\\ar=\\int_{\\mathbf{G}(S_1)} f(\\mathbf{X})\\,d\\mathbf{X}-\\int_{S_1}", " f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\\\", "\\ar{}+\\sum_{j=1}^r\\int_{\\mathbf{G}(C_j)}(f(\\mathbf{X})-", "f(\\mathbf{A}_j))\\,d\\mathbf{X}\\\\", "\\ar{}+\\sum_{j=1}^r\\int_{C_j}((f(\\mathbf{G}(\\mathbf{Y}_j))-", "f(\\mathbf{G}(\\mathbf{Y})))|J(\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\\\", "\\ar{}+\\sum_{j=1}^r f(\\mathbf{X}_j)\\left(V(\\mathbf{G}(C_j))-", "\\int_{C_j} |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\right).", "\\end{eqnarray*}", "\\newpage", "\\noindent", "Since $f(\\mathbf{X})\\ge0$,", "$$", "\\int_{S_1}f(\\mathbf{G}(\\mathbf{Y}))|J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\ge0,", "$$", "and", "Lemma~\\ref{thmtype:7.3.11}", "implies that the last", "sum is nonpositive.", "Therefore,", "\\begin{equation} \\label{eq:7.3.42}", "Q(S)\\le I_1+I_2+I_3,", "\\end{equation}", "where", "$$", "I_1=\\int_{\\mathbf{G}(S_1)} f(\\mathbf{X})\\,d\\mathbf{X},\\quad", "I_2=", "\\sum_{j=1}^r\\int_{\\mathbf{G}(C_j)}|f(\\mathbf{X})-f(\\mathbf{X}_j)|", "\\,d\\mathbf{X},", "$$", "and", "$$", "I_3=", "\\sum_{j=1}^r\\int_{C_j}|f(\\mathbf{G})(\\mathbf{Y}_j))-f(\\mathbf{G}(\\mathbf{Y}))|", " |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}.", "$$", "We will now estimate these three terms. Suppose that $\\epsilon>0$.", "To estimate $I_1$, we first remind you that since $\\mathbf{G}$", "is regular on the compact set $S$, $\\mathbf{G}$ is also regular on some", "open", "set ${\\mathcal O}$ containing $S$ (Definition~\\ref{thmtype:6.3.2}).", "Therefore, since $S_1\\subset S$ and $V(S_1)<\\epsilon$,", "$S_1$ can be covered by cubes $T_1$, $T_2$, \\dots, $T_m$ such that", "\\begin{equation} \\label{eq:7.3.43}", "\\sum_{j=1}^r V(T_j)< \\epsilon", "\\end{equation}", " and $\\mathbf{G}$ is regular on $\\bigcup_{j=1}^m", "T_j$. Now,", "$$", "\\begin{array}{rcll}", "I_1\\ar\\le M_2V(\\mathbf{G}(S_1))& \\mbox{(from", "\\eqref{eq:7.3.39})}\\\\[2\\jot]", "\\ar\\le M_2\\dst\\sum_{j=1}^m V(\\mathbf{G}(T_j))&(\\mbox{since", "}S_1\\subset\\cup_{j=1}^mT_j)\\\\[2\\jot]", "\\ar\\le M_2\\dst\\sum_{j=1}^m\\int_{T_j}| J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}&", "\\mbox{(from Lemma~\\ref{thmtype:7.3.11})}", "\\\\[2\\jot]", "\\ar\\le M_2M_1\\epsilon& \\mbox{(from \\eqref{eq:7.3.38}", "and", "\\eqref{eq:7.3.43})}.", "\\end{array}", "$$", "To estimate $I_2$, we note that", "if $\\mathbf{X}$ and $\\mathbf{X}_j$ are in $\\mathbf{G}(C_j)$", "then $\\mathbf{X}=\\mathbf{G}(\\mathbf{Y})$ and", "$\\mathbf{X}_j=\\mathbf{G}(\\mathbf{Y}_j)$ for some $\\mathbf{Y}$ and $\\mathbf{Y}_j$ in", "$C_j$. Since the edge length of $C_j$ is less than", "$\\delta/\\sqrt n$, it follows that $|\\mathbf{Y}-\\mathbf{Y}_j|<\\delta$, so", " $|f(\\mathbf{X})-f(\\mathbf{X}_j)|<\\epsilon$, by \\eqref{eq:7.3.40}.", "Therefore,", "$$", "\\begin{array}{rcll}", "I_2\\ar< \\epsilon\\dst\\sum_{j=1}^r V(\\mathbf{G}(C_j))\\\\[2\\jot]", "\\ar\\le \\epsilon\\dst\\sum_{j=1}^r\\int_{C_j}|J\\mathbf{G}(\\mathbf{Y})|d\\mathbf{Y}&", "\\mbox{(from Lemma~\\ref{thmtype:7.3.11})}\\\\[2\\jot]", "\\ar\\le \\dst\\epsilon M_1\\sum_{j=1}^r V(C_j)&\\mbox{(from", "\\eqref{eq:7.3.38}})\\\\[2\\jot]", "\\ar\\le \\epsilon M_1 V(S)&(\\mbox{since }\\dst\\cup_{j=1}^rC_j\\subset S).", "\\end{array}", "$$", "\\newpage", "To estimate $I_3$, we note again from \\eqref{eq:7.3.40} that", " $|f(\\mathbf{G}(\\mathbf{Y}_j))-f(\\mathbf{G}(\\mathbf{Y}))|<", " \\epsilon$ if $\\mathbf{Y}$ and $\\mathbf{Y}_j$ are in $C_j$.", " Hence,", "\\begin{eqnarray*}", "I_3\\ar< \\epsilon\\sum_{j=1}^r", "\\int_{C_j}|J\\mathbf{G}(\\mathbf{Y})|d\\mathbf{Y}\\\\", "\\ar\\le M_1\\epsilon\\sum_{j=1}^r V(C_j)", "\\mbox{\\quad(from \\eqref{eq:7.3.38}}\\\\", "\\ar\\le M_1 V(S)\\epsilon", "\\end{eqnarray*}", "because $\\bigcup_{j=1}^r C_j\\subset S$ and $C_i^0\\cap C_j^0=\\emptyset$", "if", "$i\\ne j$.", "From these inequalities on $I_1$, $I_2$, and $I_3$,", "\\eqref{eq:7.3.42} now implies that", "$$", "Q(S)1$ and", "$q=p/(p-1);$ thus$,$", "\\begin{equation} \\label{eq:8.1.5}", "\\frac{1}{p}+\\frac{1}{q}=1.", "\\end{equation}", " Then", "\\begin{equation} \\label{eq:8.1.6}", "\\sum_{i=1}^n \\mu_i\\nu_i\\le\\left(\\sum_{i=1}^n\\mu_i^p\\right)^{1/p}", "\\left(\\sum_{i=1}^n \\nu_i^q\\right)^{1/q}.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Let $\\alpha$ and $\\beta$ be any two positive numbers, and", "consider the function", "$$", "f(\\beta)=\\frac{\\alpha^p}{p}+\\frac{\\beta^q}{q}-\\alpha\\beta,", "$$", "\\newpage", "\\noindent", "where we regard $\\alpha$ as a constant. Since $f'(\\beta)=\\beta^{q-1}-\\alpha$ and", "$f''(\\beta)=(q-1)\\beta^{q-2}>0$ for $\\beta>0$, $f$ assumes its minimum value", "on $[0,\\infty)$ at $\\beta=\\alpha^{1/(q-1)}=\\alpha^{p-1}$. But", "$$", "f(\\alpha^{p-1})=\\frac{\\alpha^p}{p}+\\frac{\\alpha^{(p-1)q}}{q}-\\alpha^p", "=\\alpha^p\\left(\\frac{1}{p}+\\frac{1}{q}-1\\right)=0.", "$$", "Therefore,", "\\begin{equation} \\label{eq:8.1.7}", "\\alpha\\beta\\le \\frac{\\alpha^p}{p}+\\frac{\\beta^q}{q}\\mbox{\\quad if \\quad}", "\\alpha, \\beta\\ge0.", "\\end{equation}", "Now let", "$$", "\\alpha_i=\\mu_i\\left(\\sum_{j=1}^n \\mu_j^p\\right)^{-1/p}", "\\mbox{\\quad and \\quad}", "\\beta_i=\\nu_i\\left(\\sum_{j=1}^n \\nu_j^q\\right)^{-1/q}.", "$$", "From \\eqref{eq:8.1.7},", "$$", "\\alpha_i\\beta_i\\le\\frac{\\mu_i^p}{p}\\left(\\sum_{j=1}^n \\mu_j^p\\right)^{-1}", "+\\frac{\\nu_i^q}{q}\\left(\\sum_{j=1}^n \\nu_j^q\\right)^{-1}.", "$$", "From \\eqref{eq:8.1.5}, summing this from $i=1$ to $n$ yields $\\sum_{i=1}^n", "\\alpha_i\\beta_i\\le1$, which implies", "\\eqref{eq:8.1.6}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 272, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.8", "categories": [], "title": "", "contents": [ "Suppose that $u_1,$ $u_2,$ \\dots$,$ $u_n$ and $v_1,$ $v_2,$ \\dots$,$ $v_n$", "are nonnegative numbers and $p>1.$ Then", "\\begin{equation} \\label{eq:8.1.8}", "\\left(\\sum_{i=1}^n(u_i+v_i)^p\\right)^{1/p}", "\\le\\left(\\sum_{i=1}^n u_i^p\\right)^{1/p}", "+\\left(\\sum_{i=1}^n v_i^p\\right)^{1/p}.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Again, let $q=p/(p-1)$. We write", "\\begin{equation} \\label{eq:8.1.9}", "\\sum_{i=1}^n(u_i+v_i)^p=\\sum_{i=1}^n u_i(u_i+v_i)^{p-1}", "+\\sum_{i=1}^n v_i(u_i+v_i)^{p-1}.", "\\end{equation}", "From H\\\"older's inequality with $\\mu_i=u_i$ and", "$\\nu_i=(u_i+v_i)^{p-1}$,", "\\begin{equation} \\label{eq:8.1.10}", "\\sum_{i=1}^n u_i(u_i+v_i)^{p-1}\\le", "\\left(\\sum_{i=1}^n u_i^p\\right)^{1/p}", "\\left(\\sum_{i=1}^n(u_i+v_i)^p\\right)^{1/q},", "\\end{equation}", "since $q(p-1)=p$. Similarly,", "$$", "\\sum_{i=1}^n v_i(u_i+v_i)^{p-1}\\le", "\\left(\\sum_{i=1}^n v_i^p\\right)^{1/p}", "\\left(\\sum_{i=1}^n(u_i+v_i)^p\\right)^{1/q}.", "$$", "This, \\eqref{eq:8.1.9}, and \\eqref{eq:8.1.10} imply that", "$$", "\\sum_{i=1}^n(u_i+v_i)^p", "\\le\\left[\\left(\\sum_{i=1}^n u_i^p\\right)^{1/p}", "+\\left(\\sum_{i=1}^n v_i^p\\right)^{1/p}\\right]", "\\left(\\sum_{i=1}^n(u_i+v_i)^p\\right)^{1/q}.", "$$", "\\newpage", "\\noindent", "Since $1-1/q=1/p$, this implies \\eqref{eq:8.1.8}, which is", "known as {\\it Minkowski's inequality\\/}." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 273, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.2", "categories": [], "title": "", "contents": [ "If $a$ and $b$ are any two real numbers$,$ then", "\\begin{equation} \\label{eq:1.1.4}", "|a-b|\\ge\\big||a|-|b|\\big|", "\\end{equation}", "and", "\\begin{equation} \\label{eq:1.1.5}", "|a+b|\\ge\\big||a|-|b|\\big|.", "\\end{equation}" ], "refs": [], "proofs": [ { "contents": [ "Replacing $a$ by $a-b$ in \\eqref{eq:1.1.3} yields", "$$", "|a|\\le|a-b|+|b|,", "$$", "so", "\\begin{equation} \\label{eq:1.1.6}", "|a-b|\\ge|a|-|b|.", "\\end{equation}", "Interchanging $a$ and $b$ here yields", "$$", "|b-a|\\ge|b|-|a|,", "$$", "which is equivalent to", "\\begin{equation} \\label{eq:1.1.7}", "|a-b|\\ge|b|-|a|,", "\\end{equation}", "since $|b-a|=|a-b|$. Since", "$$", "\\big||a|-|b|\\big|=", "\\left\\{\\casespace\\begin{array}{l} |a|-|b|\\mbox{\\quad if \\quad} |a|>|b|,\\\\[2\\jot]", " |b|-|a|\\mbox{\\quad if \\quad} |b|>|a|,", "\\end{array}\\right.", "$$", "\\eqref{eq:1.1.6} and \\eqref{eq:1.1.7} imply \\eqref{eq:1.1.4}. Replacing", "$b$ by $-b$ in \\eqref{eq:1.1.4} yields \\eqref{eq:1.1.5}, since", "$|-b|=|b|$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 274, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.6", "categories": [], "title": "", "contents": [ "contains all its limit points$.$" ], "refs": [], "proofs": [ { "contents": [ "Since $S$ is bounded, it has an infimum $\\alpha$", "and a supremum $\\beta$, and, since $S$ is closed, $\\alpha$", "and $\\beta$ belong to $S$ (Exercise~\\ref{exer:1.3.17}). Define", "$$", "S_t=S\\cap [\\alpha,t] \\mbox{\\quad for \\ } t\\ge\\alpha,", "$$", "and let", "$$", "F=\\set{t}{\\alpha\\le t\\le\\beta \\mbox{\\ and finitely many sets from", "${\\mathcal H}$ cover $S_t$}}.", "$$", "Since $S_\\beta=S$, the theorem will be proved if we can show that", "$\\beta", "\\in F$. To do this, we use the completeness of the reals.", "Since $\\alpha\\in S$, $S_\\alpha$ is the singleton set $\\{\\alpha\\}$,", "which is contained in some open set $H_\\alpha$ from ${\\mathcal H}$", "because ${\\mathcal H}$ covers $S$; therefore, $\\alpha\\in F$. Since $F$ is", "nonempty and bounded above by $\\beta$, it has a supremum $\\gamma$.", "First, we wish to show that $\\gamma=\\beta$. Since $\\gamma\\le\\beta$ by", "definition of $F$, it suffices to rule out the possibility that", "$\\gamma<\\beta$. We consider two cases.", "{\\sc Case 1}. Suppose that $\\gamma<\\beta$ and $\\gamma\\not\\in S$. Then,", "since $S$ is closed, $\\gamma$ is not a limit point of $S$", "(Theorem~\\ref{thmtype:1.3.5}). Consequently, there is an $\\epsilon>0$", "such that", "$$", "[\\gamma-\\epsilon,\\gamma+\\epsilon]\\cap S=\\emptyset,", "$$", "so $S_{\\gamma-\\epsilon}=S_{\\gamma+\\epsilon}$. However, the", "definition of $\\gamma$ implies that $S_{\\gamma-\\epsilon}$ has a finite", "subcovering from ${\\mathcal H}$, while $S_{\\gamma+\\epsilon}$ does not.", "This is a contradiction.", "{\\sc Case 2}. Suppose that $\\gamma<\\beta$ and $\\gamma\\in S$. Then", "there is an open", "set $H_\\gamma$ in ${\\mathcal H}$ that contains $\\gamma$ and, along with $\\gamma$, an", "interval $[\\gamma-\\epsilon,\\gamma+\\epsilon]$ for some positive", "$\\epsilon$.", "Since $S_{\\gamma-\\epsilon}$ has a finite covering $\\{H_1, \\dots,H_n\\}$ of", "sets from ${\\mathcal H}$, it follows that $S_{\\gamma+\\epsilon}$ has the finite", "covering $\\{H_1, \\dots,H_n,H_\\gamma\\}$. This contradicts the", "definition of $\\gamma$.", "Now we know that $\\gamma=\\beta$, which is in $S$. Therefore, there is", "an open set $H_\\beta$ in ${\\mathcal H}$ that contains $\\beta$ and along", "with $\\beta$, an interval of the form", "$[\\beta-\\epsilon,\\beta+\\epsilon]$, for some positive $\\epsilon$. Since", "$S_{\\beta-\\epsilon}$ is covered by a finite collection of sets", "$\\{H_1, \\dots,H_k\\}$, $S_\\beta$ is covered by the finite collection", "$\\{H_1, \\dots, H_k, H_\\beta\\}$. Since $S_\\beta=S$, we are", "finished." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:1.3.5" ], "ref_ids": [ 10 ] } ], "ref_ids": [] }, { "id": 275, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.13", "categories": [], "title": "", "contents": [ "If $f$ is continuous on a set $T,$ then $f$ is uniformly continuous", "on any finite closed interval contained in $T.$" ], "refs": [], "proofs": [ { "contents": [ "We assume that $f$ is nondecreasing, and", "leave the case where $f$ is nonincreasing to you", "(Exercise~\\ref{exer:2.2.34}).", "Theorem~\\ref{thmtype:2.1.9}\\part{a}", "implies that the set $\\widetilde R_f=\\set{f(x)}{x\\in(a,b)}$", "is a subset of the open interval $(f(a+),f(b-))$. Therefore,", "\\begin{equation} \\label{eq:2.2.16}", "R_f=\\{f(a)\\}\\cup\\widetilde", "R_f\\cup\\{f(b)\\}\\subset\\{f(a)\\}\\cup(f(a+),f(b-))\\cup\\{f(b)\\}.", "\\end{equation}", "Now", "suppose that $f$ is continuous on $[a,b]$. Then $f(a)=f(a+)$,", "$f(b-)=f(b)$,", "so \\eqref{eq:2.2.16} implies that", "$R_f\\subset[f(a),f(b)]$. If $f(a)<\\mu0$ there is an integer", "$K$ such that", "$$", "\\left|\\sum_{n=k}^\\infty a_n\\right|<\\epsilon\\mbox{\\quad if\\quad} k\\ge", "K;", "$$", "that is$,$", "$$", "\\lim_{k\\to\\infty}\\sum_{n=k}^\\infty a_n=0.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Since $A_n=A_{n-1}+a_n$ and $a_n\\ge0$ $(n\\ge k)$, the sequence", "$\\{A_n\\}$ is nondecreasing, so the conclusion follows from", "Theorem~\\ref{thmtype:4.1.6}\\part{a} and", "Definition~\\ref{thmtype:4.3.1}.", "\\newline\\mbox{}" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.1.6", "TRENCH_REAL_ANALYSIS-thmtype:4.3.1" ], "ref_ids": [ 83, 329 ] } ], "ref_ids": [] }, { "id": 279, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.12", "categories": [], "title": "", "contents": [ "Suppose that $a_n\\ge0$ and $b_n>0$ for $n\\ge k,$ and", "$$", "\\lim_{n\\to\\infty}\\frac{a_n}{ b_n}=L,", "$$", "where $00\\ (n\\ge k)$ and", "$$", "\\lim_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}=L.", "$$", "\\vskip-1em", "Then", "\\begin{alist}", "\\item % (a)", " $\\sum a_n<\\infty$ if $L<1.$", "\\item % (b)", " $\\sum a_n=\\infty$ if $L>1.$", "\\end{alist}", "The test is inconclusive if $L=1.$" ], "refs": [], "proofs": [ { "contents": [ "\\part{a}", "We need the inequality", "\\begin{equation}\\label{eq:4.3.15}", "\\frac{1}{(1+x)^p}>1-px,\\quad x>0,\\ p>0.", "\\end{equation}", "This follows from Taylor's theorem", "(Theorem~\\ref{thmtype:2.5.4}), which implies that", "$$", "\\frac{1}{(1+x)^p}=1-px+\\frac{1}{2}\\frac{p(p+1)}{(1+c)^{p+2}}x^2,", "$$", "where $00$,", "this implies \\eqref{eq:4.3.15}.", "Now suppose that $M<-p<-1$. Then there is an integer $k$ such that", "$$", "n\\left(\\frac{a_{n+1}}{ a_n}-1\\right)<-p,\\quad n\\ge k,", "$$", "so", "$$", "\\frac{a_{n+1}}{ a_n}<1-\\frac{p}{ n},\\quad n\\ge k.", "$$", "Hence,", "$$", "\\frac{a_{n+1}}{ a_n}<\\frac{1}{(1+1/n)^p},\\quad n\\ge k,", "$$", "as can be seen by letting $x=1/n$ in \\eqref{eq:4.3.15}. From this,", "$$", "\\frac{a_{n+1}}{ a_n}<\\frac{1}{(n+1)^p}\\bigg/\\frac{1}{ n^p},\\quad n\\ge k.", "$$", " Since $\\sum 1/n^p<\\infty$ if $p>1$,", " Theorem~\\ref{thmtype:4.3.13}\\part{a} implies that", " $\\sum a_n<\\infty$.", "\\part{b} Here we need the inequality", "\\begin{equation}\\label{eq:4.3.16}", "(1-x)^q<1-qx,\\quad 0-q,\\quad n\\ge k,", "$$", "so", "$$", "\\frac{a_{n+1}}{ a_n}\\ge1-\\frac{q}{ n},\\quad n\\ge k.", "$$", "If $q\\le0$, then $\\sum a_n=\\infty$, by Corollary~\\ref{thmtype:4.3.6}.", "Hence, we may assume that $0\\left(1-\\frac{1}{ n}\\right)^q,\\quad n\\ge k,", "$$", "\\newpage", "\\noindent", "as can be seen by setting $x=1/n$ in \\eqref{eq:4.3.16}. Hence,", "$$", "\\frac{a_{n+1}}{ a_n}>\\frac{1}{ n^q}\\bigg/\\frac{1}{(n-1)^q},\\quad n\\ge k.", "$$", " Since $\\sum 1/n^q=\\infty$ if $q<1$,", " Theorem~\\ref{thmtype:4.3.13}\\part{b} implies that", " $\\sum a_n=\\infty$." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:2.5.4", "TRENCH_REAL_ANALYSIS-thmtype:4.3.13", "TRENCH_REAL_ANALYSIS-thmtype:4.3.6", "TRENCH_REAL_ANALYSIS-thmtype:4.3.13" ], "ref_ids": [ 42, 103, 277, 103 ] } ], "ref_ids": [] }, { "id": 281, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.21", "categories": [], "title": "", "contents": [ "The series $\\sum a_nb_n$ converges if $a_{n+1}\\le a_n$ for $n\\ge k,$", "$\\lim_{n\\to\\infty}a_n=0,$ and", "$$", "|b_k+b_{k+1}+\\cdots+b_n|\\le M,\\quad n\\ge k,", "$$", "for some constant $M.$" ], "refs": [], "proofs": [ { "contents": [ "If $a_{n+1}\\le a_n$, then", "$$", "\\sum_{n=k}^m |a_{n+1}-a_n|=\\sum_{n=k}^m (a_n-a_{n+1})=a_k-a_{m+1}.", "$$", "Since $\\lim_{m\\to\\infty} a_{m+1}=0$, it follows that", "$$", "\\sum_{n=k}^\\infty |a_{n+1}-a_n|=a_k<\\infty.", "$$", "Therefore, the hypotheses of Dirichlet's test are satisfied,", "so $\\sum a_nb_n$ converges." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 282, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.22", "categories": [], "title": "Alternating Series Test", "contents": [ "The series $\\sum (-1)^na_n$ converges if $0\\le a_{n+1}\\le a_n$ and", "$\\lim_{n\\to\\infty} a_n=0.$" ], "refs": [], "proofs": [ { "contents": [ "Let $b_n=(-1)^n$; then $\\{|B_n|\\}$ is a sequence of zeros and", "ones and therefore bounded. The conclusion now follows from", "Abel's test." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 283, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.8", "categories": [], "title": "", "contents": [ "If $\\{F_n\\}$ converges uniformly to $F$ on $S$ and each $F_n$ is", "continuous on $S,$ then so is $F;$ that is$,$ a uniform limit of", "continuous functions is continuous." ], "refs": [], "proofs": [ { "contents": [ "Since", "\\begin{eqnarray*}", "\\left|\\int_a^b F_n(x)\\,dx-\\int_a^b F(x)\\,dx\\right|\\ar\\le \\int_a^b", "|F_n(x)-F(x)|\\,dx\\\\", "\\ar\\le (b-a)\\|F_n-F\\|_S", "\\end{eqnarray*}", "and $\\lim_{n\\to\\infty}\\|F_n-F\\|_S=0$, the conclusion follows." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 284, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.14", "categories": [], "title": "", "contents": [ "If $\\sum f_n$ converges uniformly on $S,$ then", "$\\lim_{n\\to\\infty}\\|f_n\\|_S=0.$" ], "refs": [], "proofs": [ { "contents": [ "From Cauchy's convergence criterion for series of constants,", "there is for each $\\epsilon>0$ an integer $N$ such that", "$$", "M_n+M_{n+1}+\\cdots+M_m<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N,", "$$", "which, because of \\eqref{eq:4.4.17}, implies that", "$$", "\\|f_n\\|_S+\\|f_{n+1}\\|_S+\\cdots+\\|f_m\\|_S<\\epsilon\\mbox{\\quad if\\quad}", " m, n\\ge N.", "$$", " Lemma~\\ref{thmtype:4.4.2} and Theorem~\\ref{thmtype:4.4.13} imply that", "$\\sum f_n$ converges uniformly on $S$.", "\\mbox{}" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.4.2", "TRENCH_REAL_ANALYSIS-thmtype:4.4.13" ], "ref_ids": [ 251, 122 ] } ], "ref_ids": [] }, { "id": 285, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.17", "categories": [], "title": "", "contents": [ "The series $\\sum_{n=k}^\\infty f_ng_n$ converges uniformly on $S$ if", "$$", "f_{n+1}(x)\\le f_n(x),\\quad x\\in S,\\quad n\\ge k,", "$$", "$\\{f_n\\}$ converges uniformly to zero on $S,$ and", "$$", "\\|g_k+g_{k+1}+\\cdots+g_n\\|_S\\le M,\\quad n\\ge k,", "$$", "for some constant $M.$" ], "refs": [], "proofs": [ { "contents": [ "In any case, the series \\eqref{eq:4.5.1} converges to $a_0$ if", "$x=x_0$. If", "\\begin{equation}\\label{eq:4.5.3}", "\\sum |a_n|r^n<\\infty", "\\end{equation}", "for some $r>0$, then $\\sum a_n (x-x_0)^n$ converges", "absolutely uniformly in $[x_0-r, x_0+r]$, by Weierstrass's test", "(Theorem~\\ref{thmtype:4.4.15}) and", "Exercise~\\ref{exer:4.4.21}. From Cauchy's root test", "(Theorem~\\ref{thmtype:4.3.17}),", "\\eqref{eq:4.5.3} holds if", "$$", "\\limsup_{n\\to\\infty} (|a_n|r^n)^{1/n}<1,", "$$", "which is equivalent to", " $$", " r\\,\\limsup_{n\\to\\infty} |a_n|^{1/n}<1", "$$", "(Exercise~\\ref{exer:4.1.30}\\part{a}).", "From \\eqref{eq:4.5.2}, this can be rewritten as $rR$, then", "\\newpage", "$$", "\\frac{1}{ R}>\\frac{1}{ |x-x_0|},", "$$", "so \\eqref{eq:4.5.2} implies that", "$$", "|a_n|^{1/n}\\ge\\frac{1}{ |x-x_0|}\\mbox{\\quad and therefore\\quad}", "|a_n(x-x_0)^n|\\ge1", "$$", "for infinitely many values of $n$. Therefore, $\\sum a_n(x-x_0)^n$", "diverges (Corollary~\\ref{thmtype:4.3.6}) if $|x-x_0|>R$.", "In particular, the series diverges for all $x\\ne x_0$ if $R=0$.", "To prove the assertions concerning the possibilities at $x=x_0+R$", "and $x=x_0-R$ requires examples, which follow. (Also, see", "Exercise~\\ref{exer:4.5.1}.)" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:4.4.15", "TRENCH_REAL_ANALYSIS-thmtype:4.3.17", "TRENCH_REAL_ANALYSIS-thmtype:4.3.6" ], "ref_ids": [ 123, 106, 277 ] } ], "ref_ids": [] }, { "id": 286, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.5.6", "categories": [], "title": "", "contents": [ "If", "$$", "f(x)=\\sum^\\infty_{n=0} a_n(x-x_0)^n,\\quad |x-x_0|0;$ $(x_0,y_0)$", "is a local minimum point if $f_{xx}(x_0,y_0)>0$, or a local maximum", "point if", "$f_{xx}(x_0,y_0)<0.$", "\\item % (b)", "$(x_0,y_0)$ is not a local extreme point of $f$ if $D<0.$", "\\end{alist}" ], "refs": [], "proofs": [ { "contents": [ "Write $(x-x_0,y-y_0)=(u,v)$ and", "$$", "p(u,v)=(d^{(2)}_{\\mathbf{X}_0}f)(u,v)=Au^2+2Buv+Cv^2,", "$$", "where $A=f_{xx}(x_0,y_0)$, $B=f_{xy}(x_0,y_0)$, and", "$C=f_{yy}(x_0,y_0)$, so", "$$", "D=AC-B^2.", "$$", "If $D>0$, then $A\\ne0$, and we can write", "\\begin{eqnarray*}", "p(u,v)\\ar=A\\left(u^2+\\frac{2B}{ A} uv+\\frac{B^2}{", "A^2}v^2\\right)+\\left(C-\\frac{B^2}{ A}\\right)v^2\\\\", "\\ar=A\\left(u+\\frac{B}{ A}v\\right)^2+\\frac{D}{ A}v^2.", "\\end{eqnarray*}", "This cannot vanish unless $u=v=0$. Hence, $d^{(2)}_{\\mathbf{X}_0}f$ is", "positive definite if $A>0$ or negative definite if $A<0$, and", "Theorem~\\ref{thmtype:5.4.10}\\part{b} implies \\part{a}.", "If $D<0$, there are three possibilities:", "\\newpage", "\\begin{description}", "\\item{\\bf 1.} $A\\ne0$; then $p(1,0)=A$ and", "$\\dst{p\\left(-\\frac{B}{ A},1\\right)=\\frac{D}{ A}}$.", "\\vspace*{6pt}", "\\item{\\bf 2.} $C\\ne0$; then $p(0,1)=C$ and $\\dst{p\\left(1,", "-\\frac{B}{ C}\\right)=\\frac{D}{ C}}$.", "\\vspace*{6pt}", "\\item{\\bf 3.} $A=C=0$; then $B\\ne0$ and $p(1,1)=2B$ and $p(1,-1)=-2B$.", "\\end{description}", "In each case the two given values of $p$ differ in sign,", " so $\\mathbf{X}_0$ is not a local extreme point of $f$, from", "Theorem~\\ref{thmtype:5.4.10}\\part{a}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:5.4.10", "TRENCH_REAL_ANALYSIS-thmtype:5.4.10" ], "ref_ids": [ 167, 167 ] } ], "ref_ids": [] }, { "id": 293, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.3.5", "categories": [], "title": "", "contents": [ "If $\\mathbf{F}$ is continuously differentiable on a", "neighborhood of $\\mathbf{X}_0$ and $J\\mathbf{F}(\\mathbf{X}_0)\\ne 0,$ then", "there is an open neighborhood $N$ of $\\mathbf{X}_0$ on which the", "conclusions of Theorem~$\\ref{thmtype:6.3.4}$ hold$.$" ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:6.3.4" ], "proofs": [ { "contents": [ "By continuity, since $J\\mathbf{F}'(\\mathbf{X}_0)\\ne0$,", " $J\\mathbf{F}'(\\mathbf{X})$", " is nonzero for all $\\mathbf{X}$ in some open neighborhood $S$ of", "$\\mathbf{X}_0$. Now apply Theorem~\\ref{thmtype:6.3.4}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:6.3.4" ], "ref_ids": [ 187 ] } ], "ref_ids": [ 187 ] }, { "id": 294, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.4.2", "categories": [], "title": "", "contents": [ "Suppose that $f:\\R^{n+1}\\to \\R$ is continuously", "differentiable on an open set containing $(\\mathbf{X}_0,u_0),$ with", "$f(\\mathbf{X}_0,u_0)=0$", "and", "$f_u(\\mathbf{X}_0,u_0)\\ne0$.", "Then there is a neighborhood $M$ of $(\\mathbf{X}_0,u_0),$ contained in", "$S,$ and a neighborhood $N$ of $\\mathbf{X}_0$ in $\\R^n$ on which", "is defined a unique continuously differentiable function", "$u=u(\\mathbf{X}):\\R^n\\to", "\\R$ such that", "$$", "(\\mathbf{X},u(\\mathbf{X}))\\in M\\mbox{\\quad and \\quad}", " f_u(\\mathbf{X},u(\\mathbf{X}))\\ne0,\\quad\\mathbf{X}\\in N,", "$$", "$$", "u(\\mathbf{X}_0)=u_0, \\mbox{\\quad and \\quad}", "f(\\mathbf{X},u(\\mathbf{X}))=0,\\quad\\mathbf{X}\\in N.", "$$", "The partial derivatives of $u$ are given by", "$$", "u_{x_i}(\\mathbf{X})=-\\frac{f_{x_i}(\\mathbf{X},u(\\mathbf{X}))}{", "f_u(\\mathbf{X},u(\\mathbf{X}))},\\quad 1\\le i\\le n.", "$$" ], "refs": [], "proofs": [ { "contents": [ "We will show that if $f$ is unbounded on $R$, ${\\bf", "P}=\\{R_1,R_2, \\dots,R_k\\}$ is", "any partition of $R$, and $M>0$, then there are Riemann sums $\\sigma$", "and $\\sigma'$ of $f$ over ${\\bf P}$ such that", "\\begin{equation} \\label{eq:7.1.11}", "|\\sigma-\\sigma'|\\ge M.", "\\end{equation}", "This implies that", "$f$ cannot satisfy Definition~\\ref{thmtype:7.1.2}. (Why?)", "Let", "$$", "\\sigma=\\sum_{j=1}^kf(\\mathbf{X}_j)V(R_j)", "$$", "be a Riemann sum of $f$ over ${\\bf P}$. There must be", "an integer $i$ in $\\{1,2, \\dots,k\\}$ such that", "\\begin{equation} \\label{eq:7.1.12}", "|f(\\mathbf{X})-f(\\mathbf{X}_i)|\\ge\\frac{M }{ V(R_i)}", "\\end{equation}", "for some $\\mathbf{X}$ in $R_i$, because if this were not so, we", "would have", "$$", "|f(\\mathbf{X})-f(\\mathbf{X}_j)|<\\frac{M}{ V(R_j)},\\quad \\mathbf{X}\\in R_j,\\quad", "\\quad 1\\le j\\le k.", "$$", "If this is so, then", "\\begin{eqnarray*}", "|f(\\mathbf{X})|\\ar=|f(\\mathbf{X}_j)+f(\\mathbf{X})-f(\\mathbf{X}_j)|\\le|f(\\mathbf{X}_j)|+|f(\\mathbf{X})-f(\\mathbf{X}_j)|\\\\", "\\ar\\le |f(\\mathbf{X}_j)|+\\frac{M}{ V(R_j)},\\quad \\mathbf{X}\\in R_j,\\quad", "1\\le j\\le k.", "\\end{eqnarray*}", "However, this implies that", "$$", "|f(\\mathbf{X})|\\le\\max\\set{|f(\\mathbf{X}_j)|+\\frac{M}{ V(R_j)}}{1\\le j\\le k},", "\\quad \\mathbf{X}\\in R,", "$$", "which contradicts the assumption that $f$ is unbounded on $R$.", " Now suppose that $\\mathbf{X}$ satisfies \\eqref{eq:7.1.12}, and", "consider the Riemann sum", "$$", "\\sigma'=\\sum_{j=1}^nf(\\mathbf{X}_j')V(R_j)", "$$", "over the same partition ${\\bf P}$, where", "$$", "\\mathbf{X}_j'=\\left\\{\\casespace\\begin{array}{ll}", "\\mathbf{X}_j,&j \\ne i,\\\\", "\\mathbf{X},&j=i.\\end{array}\\right.", "$$", "Since", "$$", "|\\sigma-\\sigma'|=|f(\\mathbf{X})-f(\\mathbf{X}_i)|V(R_i),", "$$", "\\eqref{eq:7.1.12} implies \\eqref{eq:7.1.11}." ], "refs": [ "TRENCH_REAL_ANALYSIS-thmtype:7.1.2" ], "ref_ids": [ 359 ] } ], "ref_ids": [] }, { "id": 295, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.31", "categories": [], "title": "", "contents": [ "Suppose that", " $f$ is integrable on sets $S_1$ and $S_2$ such that $S_1\\cap S_2$", "has zero content$.$ Then $f$ is integrable on $S_1\\cup S_2,$ and", "$$", "\\int_{S_1\\cup S_2} f(\\mathbf{X})\\,d\\mathbf{X}=", "\\int_{S_1} f(\\mathbf{X})\\,d\\mathbf{X}+", "\\int_{S_2} f(\\mathbf{X})\\,d\\mathbf{X}.", "$$" ], "refs": [], "proofs": [ { "contents": [ "Let", "$$", "P_1: a=x_00$ a", "partition $\\mathbf{P}$ of $R$ such that $S_f(\\mathbf{P})-s_f(\\mathbf{P})<\\epsilon$,", "from Theorem~\\ref{thmtype:7.1.12}. Consequently, from", "\\eqref{eq:7.2.6}, there is", "a partition $P_2$ of $[c,d]$ such that", "$S_F(P_2)-s_F(P_2)<\\epsilon$,", " so $F$ is integrable on $[c,d]$, from", "Theorem~\\ref{thmtype:3.2.7}.", "It remains to verify \\eqref{eq:7.2.1}. From \\eqref{eq:7.2.4} and the", "definition of $\\int_c^dF(y)\\,dy$,", "there is for each $\\epsilon>0$ a $\\delta>0$ such that", "$$", "\\left|\\int_c^d F(y)\\,dy-\\sigma\\right|<\\epsilon\\mbox{\\quad if\\quad}", "\\|P_2\\|<\\delta;", "$$", "that is,", "$$", "\\sigma-\\epsilon<\\int_c^d F(y)\\,dy<\\sigma+\\epsilon\\mbox{\\quad if \\quad}", "\\|P_2\\|<\\delta.", "$$", "This and \\eqref{eq:7.2.5} imply that", "$$", "s_f(\\mathbf{P})-\\epsilon<\\int_c^d F(y)\\,dy0$ a", "partition $\\mathbf{P}$ of $R$ such that $S_f(\\mathbf{P})-s_f(\\mathbf{P})<\\epsilon$,", "from Theorem~\\ref{thmtype:7.1.12}. Consequently, from \\eqref{eq:7.2.11},", "there", "is a partition $\\mathbf{Q}$ of $T$ such that", "$S_{F_p}(\\mathbf{Q})-s_{F_p}(\\mathbf{Q})<\\epsilon$, so $F_p$ is integrable", "on $T$, from Theorem~\\ref{thmtype:7.1.12}.", "It remains to verify that", "\\begin{equation} \\label{eq:7.2.12}", "\\int_R f(\\mathbf{X})\\,d\\mathbf{X}=", "\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}.", "\\end{equation}", "From \\eqref{eq:7.2.9} and the definition of $\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}$, there", "is for each $\\epsilon>0$ a $\\delta>0$ such that", "$$", "\\left|\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", "-\\sigma\\right|<\\epsilon\\mbox{\\quad", "if\\quad}", "\\|\\mathbf{Q}\\|<\\delta;", "$$", "that is,", "$$", "\\sigma-\\epsilon<\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", "<\\sigma+", "\\epsilon\\mbox{\\quad if \\quad}\\|\\mathbf{Q}\\|<\\delta.", "$$", "This and \\eqref{eq:7.2.10} imply that", "$$", "s_f(\\mathbf{P})-\\epsilon<", "\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", "0$ such that if ${\\bf", "P}$ is any partition of $C$ with $\\|{\\bf P}\\|\\le\\delta$ and $\\sigma$", "is any Riemann sum of $\\psi_K$ over ${\\bf P}$, then", "\\begin{equation}\\label{eq:7.3.6}", "0\\le\\sigma\\le\\epsilon.", "\\end{equation}", "\\newpage", "\\noindent", "Now suppose that ${\\bf P}=\\{C_1,C_2,\\dots,C_k\\}$ is a partition of $C$", "into cubes with", "\\begin{equation}\\label{eq:7.3.7}", "\\|{\\bf P}\\|<\\min (\\rho,\\delta),", "\\end{equation}", "and let $C_1$, $C_2$, \\dots, $C_k$ be numbered so that $C_j\\cap K\\ne", "\\emptyset$ if $1\\le j\\le r$ and", "$C_j\\cap K=\\emptyset$ if $r+1\\le j\\le k$. Then \\eqref{eq:7.3.5} holds, and", "a typical Riemann sum of $\\psi_K$ over ${\\bf P}$ is of the form", "$$", "\\sigma=\\sum_{j=1}^r\\psi_K(\\mathbf{X}_j)V(C_j)", "$$", "with $\\mathbf{X}_j\\in C_j$, $1\\le j\\le r$. In particular, we", "can choose", "$\\mathbf{X}_j$ from $K$, so that $\\psi_K(\\mathbf{X}_j)=1$, and", "$$", "\\sigma=\\sum_{j=1}^r V(C_j).", "$$", "Now \\eqref{eq:7.3.6} and \\eqref{eq:7.3.7} imply that $C_1$, $C_2$, \\dots,", "$C_r$ have the required properties." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] } ], "definitions": [ { "id": 298, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.5", "categories": [], "title": "", "contents": [ "A set $D$ is {\\it dense in the reals\\/}", "if every open interval $(a,b)$ contains a member of $D$." ], "refs": [], "ref_ids": [] }, { "id": 299, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.1", "categories": [], "title": "", "contents": [ " Let $S$ and $T$ be sets.", "\\begin{alist}", "\\item % (a)", "$S$ {\\it contains\\/} $T$, and we write $S\\supset T$ or $T\\subset", "S$, if every member of $T$ is also in $S$. In this case, $T$ is", "a {\\it subset\\/} of $S$.", "\\item % (b)", " $S-T$ is the set of elements that are in $S$ but not in $T$.", "\\item % (c)", "$S$ {\\it equals\\/} $T$, and we write $S=T$,", "if", "$S$ contains", "$T$ and", "$T$ contains $S$; thus, $S=T$ if and only if $S$ and $T$ have the same", "members.", "\\newpage", "\\item % (d)", " $S$ {\\it strictly contains\\/} $T$", "if $S$ contains $T$ but $T$ does not contain $S$; that", "is, if every member of $T$ is also in $S$, but at least one member", "of", "$S$ is not in $T$ (Figure~\\ref{figure:1.3.1}).", "\\item % (e)", "The {\\it complement\\/} of $S$, denoted by $S^c$,", "is the set of elements in the universal set that are not in $S$.", "\\item % (f)", " The {\\it union\\/} of $S$", "and", "$T$, denoted by", "$S\\cup T$, is the set of elements in at least one of $S$ and $T$", "(Figure~\\ref{figure:1.3.1}\\part{b}).", "\\item % (g)", "The {\\it intersection\\/} of $S$ and $T$, denoted by", "$S\\,\\cap\\, T$, is the", "set of elements in both $S$ and $T$ (Figure~\\ref{figure:1.3.1}\\part{c}).", "If $S\\cap T=\\emptyset$ (the empty set), then $S$ and $T$ are", " {\\it disjoint sets\\/}", "(Figure~\\ref{figure:1.3.1}\\part{d}).", "\\item % (h)", " A set with only one member $x_0$ is a {\\it singleton", "set\\/}, denoted by", "$\\{x_0\\}$.", "\\end{alist}" ], "refs": [], "ref_ids": [] }, { "id": 300, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.2", "categories": [], "title": "", "contents": [ "If $x_0$ is a real number and $\\epsilon>0$, then the open interval", "$(x_0-\\epsilon, x_0+\\epsilon)$ is an {\\it $\\epsilon$-neighborhood\\/}", "of", "$x_0$.", "If a set $S$ contains an $\\epsilon$-neighborhood of $x_0$, then $S$ is a", "{\\it neighborhood\\/} of $x_0$, and $x_0$ is an {\\it interior point\\/} of", "$S$ (Figure~\\ref{figure:1.3.2}). The set of interior points of $S$ is the", "{\\it interior\\/} of $S$, denoted by $S^0$. If every point of $S$ is an", "interior point (that is, $S^0=S$), then $S$ is {\\it open\\/}.", " A set $S$ is \\emph{closed} if $S^c$ is open." ], "refs": [], "ref_ids": [] }, { "id": 301, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.4", "categories": [], "title": "", "contents": [ "R}$. Then", "\\begin{alist}", "\\item % (a)", " $x_0$ is a {\\it limit point\\/}", "of $S$ if every deleted neighborhood of $x_0$ contains a point of~$S$.", "\\item % (b)", "$x_0$ is a {\\it boundary point\\/} of $S$ if every neighborhood", "of $x_0$ contains at least one point in $S$ and one not in $S$. The set of", "boundary points of $S$ is the {\\it boundary\\/} of $S$, denoted by $\\partial", "S$. The {\\it closure\\/} of $S$, denoted by $\\overline{S}$, is", "$\\overline{S}=S\\cup \\partial S$.", "\\item % (c)", "$x_0$ is an \\emph{isolated point} of $S$ if $x_0\\in S$", " and there is a neighborhood of $x_0$ that contains no other point of", "$S$.", "\\item % (d)", "$x_0$ is \\emph{exterior} to $S$ if $x_0$ is in the interior of $S^c$. The", "collection of such points is the {\\it exterior\\/} of $S$.", "\\end{alist}" ], "refs": [], "ref_ids": [] }, { "id": 302, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.1", "categories": [], "title": "", "contents": [ "If $D_f\\cap D_g\\ne", "\\emptyset,$ then $f+g,$ $f-g,$ and $fg$ are defined on", "$D_f\\cap D_g$ by", "\\begin{eqnarray*}", "(f+g)(x)\\ar= f(x)+g(x),\\\\", "(f-g)(x)\\ar= f(x)-g(x),\\\\", "\\noalign{\\hbox{and}}", "(fg)(x)\\ar= f(x)g(x).", "\\end{eqnarray*}", "The quotient $f/g$ is defined by", "$$", "\\left(\\frac{f}{ g}\\right) (x)=\\frac{f(x)}{ g(x)}", "$$", "for $x$ in $D_f\\cap D_g$ such that $g(x)\\ne0.$" ], "refs": [], "ref_ids": [] }, { "id": 303, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.2", "categories": [], "title": "", "contents": [ " We say that $f(x)$ {\\it approaches the limit $L$ as $x$ approaches\\/}", "$x_0$, and write", "$$", "\\lim_{x\\to x_0} f(x)=L,", "$$", "if $f$ is defined on some deleted neighborhood of $x_0$ and, for", "every $\\epsilon>0$, there is a $\\delta>0$ such that", "\\begin{equation}\\label{eq:2.1.4}", "|f(x)-L|<\\epsilon", "\\end{equation}", "if", "\\begin{equation}\\label{eq:2.1.5}", "0<|x-x_0|<\\delta.", "\\end{equation}", "Figure~\\ref{figure:2.1.1} depicts the graph", "of a function for which", "$\\lim_{x", "\\to x_0}f(x)$ exists." ], "refs": [], "ref_ids": [] }, { "id": 304, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.5", "categories": [], "title": "", "contents": [ "\\begin{alist}", "\\item % (a)", "We say that $f(x)$ {\\it approaches the left-hand limit $L$ as", "$x$ approaches $x_0$ from the left\\/}, and write", "$$", "\\lim_{x\\to x_0-} f(x)=L,", "$$", "if $f$ is defined on some open interval $(a,x_0)$ and, for each", "$\\epsilon>0$, there is a $\\delta>0$ such that", "$$", "|f(x)-L|<\\epsilon\\mbox{\\quad if \\quad} x_0-\\delta0$, there is a $\\delta>0$ such that", "$$", "|f(x)-L|<\\epsilon\\mbox{\\quad if \\quad} x_00$, there is a number $\\beta$ such that", "$$", "|f(x)-L|<\\epsilon\\quad\\mbox{\\quad if \\quad} x>\\beta.", "\\eqno{\\bbox}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 306, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.8", "categories": [], "title": "", "contents": [ "We say that $f(x)$ {\\it approaches $\\infty$ as $x$ approaches $x_0$", "from the left\\/}, and write", "$$", "\\lim_{x\\to x_0-} f(x)=\\infty\\mbox{\\quad or \\quad} f(x_0-)=\\infty,", "$$", "if $f$ is defined on an interval $(a,x_0)$ and, for each real number", "$M$, there is a $\\delta>0$ such that", "$$", "f(x)>M\\mbox{\\quad if \\quad} x_0-\\delta0$, there is a $\\delta>0$ such", "that", "$$", "|f(x)-f(x')|<\\epsilon\\mbox{\\ whenever }\\ |x-x'|<\\delta", "\\mbox{\\ and }\\ x,x'\\in S.", "\\eqno{\\bbox}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 313, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.1", "categories": [], "title": "", "contents": [ "A function $f$ is {\\it differentiable\\/}", "at an interior point $x_0$ of its domain if the difference quotient", "$$", "\\frac{f(x)-f(x_0)}{ x-x_0},\\quad x\\ne x_0,", "$$", "approaches a limit as $x$ approaches $x_0$, in which case the limit is", "called the {\\it derivative of $f$ at $x_0$\\/}, and", "is denoted by", "$f'(x_0)$; thus,", "\\begin{equation}\\label{eq:2.3.1}", "f'(x_0)=\\lim_{x\\to x_0}\\frac{f(x)-f(x_0)}{ x-x_0}.", "\\end{equation}", "It is sometimes convenient to let $x=x_0+h$ and write \\eqref{eq:2.3.1}", "as", "$$", "f'(x_0)=\\lim_{h\\to 0}\\frac{f(x_0+h)-f(x_0)}{ h}.", "\\eqno{\\bbox}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 314, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.6", "categories": [], "title": "", "contents": [ "\\begin{alist}", "\\item % (a)", "We say that $f$ is {\\it differentiable on the closed interval\\/}", "$[a,b]$ if $f$ is differentiable on the open interval $(a,b)$ and", "$f_+'(a)$ and $f_-'(b)$ both exist.", "\\item % (b)", "We say that $f$ is {\\it continuously differentiable on\\/}", "$[a,b]$ if $f$ is differentiable on $[a,b]$, $f'$ is continuous", "on $(a,b)$,", "$f_+'(a)=f'(a+)$, and $f_-'(b)=f'(b-)$.", "\\end{alist}" ], "refs": [], "ref_ids": [] }, { "id": 315, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.1.1", "categories": [], "title": "", "contents": [ "Let $f$ be defined on $[a,b]$. We say that $f$ is", "{\\it Riemann integrable on\\/}", "$[a,b]$ if there", "is a number $L$ with the following property: For every $\\epsilon>0$,", "there is a $\\delta>0$ such that", "$$", "\\left|\\sigma-L \\right|<\\epsilon", "$$", "if $\\sigma$ is any Riemann sum of $f$ over", "a partition $P$ of $[a,b]$", "such that $\\|P\\|<\\delta$.", "In this case, we say that $L$ is {\\it the Riemann integral of", "$f$ over\\/} $[a,b]$,", "and write", "$$", "\\int_a^b f(x)\\,dx=L.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 316, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.1.3", "categories": [], "title": "", "contents": [ "If $f$ is bounded on $[a,b]$ and", "$P=\\{x_0,x_1, \\dots,x_n\\}$ is a partition of $[a,b]$, let", "\\begin{eqnarray*}", "M_j\\ar=\\sup_{x_{j-1}\\le x\\le x_j}f(x)\\\\", "\\arraytext{and}\\\\", "m_j\\ar=\\inf_{x_{j-1}\\le x\\le x_j}f(x).", "\\end{eqnarray*}", "The {\\it upper sum of $f$ over $P$\\/}", " is", "$$", "S(P)=\\sum_{j=1}^n M_j(x_j-x_{j-1}),", "$$", "and the {\\it upper integral of $f$ over\\/},", "$[a,b]$, denoted by", "$$", "\\overline{\\int_a^b} f(x)\\,dx,", "$$", "is the infimum of all upper sums. The {\\it lower", "sum of $f$ over $P$\\/}", "is", "$$", "s(P)=\\sum_{j=1}^n m_j(x_j-x_{j-1}),", "$$", "and the {\\it lower integral of $f$ over\\/}", "$[a,b]$, denoted by", "$$", "\\underline{\\int_a^b}f(x)\\,dx,", "$$", "is the supremum of all lower sums.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 317, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.1.5", "categories": [], "title": "", "contents": [ "Let $f$ and $g$ be defined on $[a,b]$. We say that $f$ is", "{\\it Riemann}--\\href{http://www-history.mcs.st-and.ac.uk/Mathematicians/Stieltjes.html}", "{\\it Stieltjes}", "{\\it integrable with respect to $g$ on\\/}", "$[a,b]$", "if there", "is a number $L$ with the following property: For every $\\epsilon>0$,", "there is a $\\delta>0$ such that", "\\begin{equation} \\label{eq:3.1.15}", "\\left|\\sum_{j=1}^n f(c_j)\\left[g(x_j)-g(x_{j-1})\\right]-L \\right|<", "\\epsilon,", "\\end{equation}", "provided only that $P=\\{x_0,x_1, \\dots,x_n\\}$ is a partition of $[a,b]$", "such that $\\|P\\|<\\delta$ and", "$$", "x_{j-1}\\le c_j\\le x_j,\\quad j=1,2, \\dots,n.", "$$", "In this case, we say that $L$ is {\\it the Riemann--Stieltjes integral", "of", "$f$ with respect to $g$ over\\/}", "$[a,b]$, and write", "$$", "\\int_a^b f(x)\\,dg (x)=L.", "$$", "The sum", "$$", "\\sum_{j=1}^n f(c_j)\\left[g(x_j)-g(x_{j-1})\\right]", "$$", "in \\eqref{eq:3.1.15} is {\\it a Riemann--Stieltjes sum of $f$", "with respect to $g$ over the partition~$P$\\/}." ], "refs": [], "ref_ids": [] }, { "id": 318, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.4.1", "categories": [], "title": "", "contents": [ "If $f$ is locally integrable on", "$[a,b)$, we define", "\\begin{equation}\\label{eq:3.4.1}", "\\int_a^b f(x)\\,dx=", "\\lim_{c\\to b-}\\int_a^c f(x)\\,dx", "\\end{equation}", "if the limit exists (finite). To include the case where $b=\\infty$, we", "adopt the convention that $\\infty-=\\infty$.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 319, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.4.2", "categories": [], "title": "", "contents": [ " If $f$ is locally integrable on", "$(a,b]$, we define", "$$", "\\int_a^b f(x)\\,dx=\\lim_{c\\to a+}\\int_c^b f(x)\\,dx", "$$", "provided that the limit exists (finite).", " To include the case where $a=-\\infty$, we adopt the", "convention that $-\\infty+=-\\infty$." ], "refs": [], "ref_ids": [] }, { "id": 320, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:3.4.3", "categories": [], "title": "", "contents": [ "If $f$ is locally integrable on", "$(a,b),$ we define", "$$", "\\int_a^b f(x)\\,dx=\\int_a^\\alpha f(x)\\,dx+\\int_\\alpha^b f(x)\\,dx,", "$$", "where $a<\\alpha0$ there is a finite or infinite sequence of", "open intervals $I_1$, $I_2$, \\dots\\ such that", "\\begin{equation} \\label{eq:3.5.8}", "S\\subset\\bigcup_j I_j", "\\end{equation}", "and", "\\begin{equation} \\label{eq:3.5.9}", "\\sum_{j=1}^n L(I_j)<\\epsilon,\\quad n\\ge1.", "\\end{equation}" ], "refs": [], "ref_ids": [] }, { "id": 324, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.1", "categories": [], "title": "", "contents": [ "A sequence $\\{s_n\\}$ {\\it converges to a limit $s$\\/} if for", "every $\\epsilon>0$ there is an integer $N$ such that", "\\begin{equation}\\label{eq:4.1.2}", "|s_n-s|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N.", "\\end{equation}", "In this case we say that $\\{s_n\\}$ is {\\it convergent\\/} and write", "$$", "\\lim_{n\\to\\infty}s_n=s.", "$$", "A sequence that does not converge {\\it diverges\\/}, or is", "{\\it divergent\\/}", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 325, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.3", "categories": [], "title": "", "contents": [ "A sequence $\\{s_n\\}$ is {\\it bounded above\\/}", " if there is a real number $b$ such that", "$$", "s_n\\le b\\mbox{\\quad for all $n$},", "$$", "{\\it bounded below\\/} if there is a", "real number", "$a$ such that", "$$", "s_n\\ge a\\mbox{\\quad for all $n$},", "$$", "or {\\it bounded\\/} if", "there is a real number", "$r$ such that", "$$", "|s_n|\\le r\\mbox{\\quad for all $n$}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 326, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.5", "categories": [], "title": "", "contents": [ " A sequence $\\{s_n\\}$ is {\\it nondecreasing\\/} if", "$s_n\\ge", "s_{n-1}$ for all $n$, or {\\it nonincreasing\\/} if", "$s_n\\le s_{n-1}$", "for all $n.$ A {\\it monotonic sequence\\/}", "is a sequence that is either", "nonincreasing or nondecreasing. If $s_n>s_{n-1}$ for all $n$, then", "$\\{s_n\\}$ is {\\it increasing\\/},", "while if", "$s_n 0$ there is an integer $N$ such that", "\\begin{equation} \\label{eq:4.4.1}", "\\|F_n-F\\|_S<\\epsilon\\mbox{\\quad if\\quad} n\\ge N.", "\\end{equation}" ], "refs": [], "ref_ids": [] }, { "id": 334, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.12", "categories": [], "title": "", "contents": [ "If $\\{f_j\\}^\\infty_k$ is a sequence of real-valued functions defined", "on a set $D$ of reals, then $\\sum_{j=k}^\\infty f_j$ is an", "{\\it infinite series\\/} (or simply a {\\it", "series\\/}) of functions on", "$D$. The {\\it partial sums of\\/},", "$\\sum_{j=k}^\\infty f_j$ are defined by", "$$", "F_n=\\sum^n_{j=k} f_j,\\quad n\\ge k.", "$$", "If $\\{F_n\\}^\\infty_k$ converges pointwise to a function $F$ on a", "subset $S$ of $D$, we say that $\\sum_{j=k}^\\infty f_j$ {\\it converges", "pointwise to the sum $F$ on\\/} $S$, and write", "$$", "F=\\sum_{j=k}^\\infty f_j,\\quad x\\in S.", "$$", "\\newpage", "\\noindent", "If $\\{F_n\\}$ converges uniformly to $F$ on $S$, we say that", "$\\sum_{j=k}^\\infty f_j$ {\\it converges uniformly to $F$ on~$S$\\/}." ], "refs": [], "ref_ids": [] }, { "id": 335, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:4.5.1", "categories": [], "title": "", "contents": [ "An infinite series of the form", "\\begin{equation}\\label{eq:4.5.1}", "\\sum^\\infty_{n=0} a_n(x-x_0)^n,", "\\end{equation}", "where $x_0$ and $a_0$, $a_1$, \\dots, are constants, is called a {\\it", "power series in $x-x_0$\\/}.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 336, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.1", "categories": [], "title": "", "contents": [ "The {\\it vector sum\\/} of", "$$", "\\mathbf{X}=(x_1,x_2, \\dots,x_n)\\mbox{\\quad and\\quad}\\mathbf{Y}=", "(y_1,y_2, \\dots,y_n)", "$$", "is", "\\begin{equation}\\label{eq:5.1.1}", "\\mathbf{X}+\\mathbf{Y}=(x_1+y_1,x_2+y_2, \\dots,x_n+y_n).", "\\end{equation}", "If $a$ is a real number, the {\\it scalar multiple of $\\mathbf{X\\/}$ by\\/}", "$a$ is", "\\begin{equation}\\label{eq:5.1.2}", "a\\mathbf{X}=(ax_1,ax_2, \\dots,ax_n).", "\\end{equation}" ], "refs": [], "ref_ids": [] }, { "id": 337, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.3", "categories": [], "title": "", "contents": [ "The {\\it length\\/} of the vector", "$\\mathbf{X}=(x_1,x_2, \\dots, x_n)$ is", "$$", "|\\mathbf{X}|=(x^2_1+x^2_2+\\cdots+x^2_n)^{1/2}.", "$$", "The {\\it distance between points $\\mathbf{X\\/}$ and\\/} $\\mathbf{Y}$ is", "$|\\mathbf{X}-\\mathbf{Y}|$; in particular, $|\\mathbf{X}|$ is the distance between", "$\\mathbf{X}$ and the origin. If $|\\mathbf{X}|=1$, then $\\mathbf{X}$ is", "a {\\it unit vector\\/}.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 338, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.4", "categories": [], "title": "", "contents": [ "The {\\it inner product\\/} $\\mathbf{X}\\cdot", "\\mathbf{Y}$ of $\\mathbf{X}=(x_1,x_2, \\dots,x_n)$ and $\\mathbf{Y}=", "(y_1,y_2, \\dots,y_n)$ is", "$$", "\\mathbf{X}\\cdot\\mathbf{Y}=x_1y_1+x_2y_2+\\cdots+x_ny_n.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 339, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.10", "categories": [], "title": "", "contents": [ "$\\mathbf{U}$ are in $\\R^n$ and $\\mathbf{U}\\ne\\mathbf{0}$. Then {\\it the", "line through $\\mathbf{X}_0$ in the direction of\\/}", "$\\mathbf{U}$ is the set of all points in $\\R^n$ of the form", "$$", "\\mathbf{X}=\\mathbf{X}_0+t\\mathbf{U},\\quad -\\infty0$, the {\\it $\\epsilon$-neighborhood of a point\\/}", "$\\mathbf{X}_{0}$ in", "$\\R^n$ is the set", "$$", "N_\\epsilon(\\mathbf{X}_0)|=\\set{\\mathbf{X}}{|\\mathbf{X}-\\mathbf{X}_0|<\\epsilon}.", "\\eqno{\\bbox}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 341, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.13", "categories": [], "title": "", "contents": [ "A sequence of points $\\{\\mathbf{X}_r\\}$ in $\\R^n$", "{\\it converges to the limit\\/} $\\overline{\\mathbf{X}}$ if", "$$", "\\lim_{r\\to\\infty} |\\mathbf{X}_r-\\overline{\\mathbf{X}}|=0.", "$$", "In this case we write", "$$", "\\lim_{r\\to\\infty}\\mathbf{X}_r=\\overline{\\mathbf{X}}.", "\\eqno{\\bbox}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 342, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.16", "categories": [], "title": "", "contents": [ "If $S$ is a nonempty subset of $\\R^n$, then", "$$", "d(S)=\\sup\\set{|\\mathbf{X}-\\mathbf{Y}|}{\\mathbf{X},\\mathbf{Y}\\in S}", "$$", "is the {\\it diameter\\/} of $S$.", "If $d(S)<\\infty,$ $S$ is {\\it bounded\\/}$;$ if", "$d(S)=\\infty$, $S$ is {\\it unbounded\\/}." ], "refs": [], "ref_ids": [] }, { "id": 343, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.19", "categories": [], "title": "", "contents": [ "A subset $S$ of $\\R^n$ is", " {\\it connected\\/} if it is impossible to represent", "$S$ as the union of two", "disjoint nonempty sets such that neither contains a limit point of the", "other; that is, if $S$ cannot be expressed as $S=A\\cup B$, where", "\\begin{equation}\\label{eq:5.1.16}", "A\\ne\\emptyset,\\quad B\\ne\\emptyset,\\quad\\overline{A}\\cap B=", "\\emptyset,\\mbox{\\quad and\\quad} A\\cap\\overline{B}=\\emptyset.", "\\end{equation}", "If $S$ can be expressed in this way, then $S$ is", "{\\it disconnected\\/}." ], "refs": [], "ref_ids": [] }, { "id": 344, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.21", "categories": [], "title": "", "contents": [ "A {\\it region\\/} $S$ in $\\R^n$ is the union of an open connected", "set", "with some, all, or none of its boundary; thus, $S^0$ is connected, and", "every point of $S$ is a limit point of $S^0$." ], "refs": [], "ref_ids": [] }, { "id": 345, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.1", "categories": [], "title": "", "contents": [ "We say that $f(\\mathbf{X})$", "{\\it approaches the limit $L$ as $\\mathbf{X\\/}$ approaches\\/} $\\mathbf{X}_0$", "and write", "$$", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=L", "$$", "if $\\mathbf{X}_0$ is a limit point of $D_f$ and, for every $\\epsilon>0$,", "there is a $\\delta>0$ such that", "$$", "|f(\\mathbf{X})-L|<\\epsilon", "$$", "for all $\\mathbf{X}$ in $D_f$ such that", "$$", "0<|\\mathbf{X}-\\mathbf{X}_0|<\\delta.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 346, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.4", "categories": [], "title": "", "contents": [ "We say that $f(\\mathbf{X})$ {\\it approaches $\\infty$ as $\\mathbf{X\\/}$", "approaches", "$\\mathbf{X}_0$\\/} and write", "$$", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=\\infty", "$$", "if $\\mathbf{X}_0$ is a limit point of $D_f$ and, for every real number", "$M$, there is a $\\delta>0$ such that", "$$", "f(\\mathbf{X})>M\\mbox{\\quad whenever\\quad} 0<|\\mathbf{X}-\\mathbf{X}_0|<\\delta", "\\mbox{\\quad and\\quad}\\mathbf{X}\\in D_f.", "$$", "We say that", "\\begin{eqnarray*}", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})\\ar=-\\infty\\\\", "\\arraytext{if}\\\\", "\\lim_{{\\mathbf{X}}\\to\\mathbf{X}_0} (-f)(\\mathbf{X})\\ar=\\infty.", "\\end{eqnarray*}" ], "refs": [], "ref_ids": [] }, { "id": 347, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.5", "categories": [], "title": "", "contents": [ "If $D_f$ is unbounded$,$ we say that", "$$", "\\lim_{|\\mathbf{X}|\\to\\infty} f(\\mathbf{X})=L\\mbox{\\quad (finite)\\quad}", "$$", "if for every $\\epsilon>0$, there is a number $R$ such that", "$$", "|f(\\mathbf{X})-L|<\\epsilon\\mbox{\\quad whenever\\quad}\\ |\\mathbf{X}|\\ge R", "\\mbox{\\quad and\\quad}\\mathbf{X}\\in D_f.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 348, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.6", "categories": [], "title": "", "contents": [ "If $\\mathbf{X}_0$ is in $D_f$ and is a limit point of $D_f$, then we say", "that $f$ is", "{\\it continuous at $\\mathbf{X\\/}_0$\\/} if", "$$", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=f(\\mathbf{X}_0).", "\\eqno{\\bbox}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 349, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.3.1", "categories": [], "title": "", "contents": [ "Let $\\boldsymbol{\\Phi}$ be a unit vector and $\\mathbf{X}$ a point in", "$\\R^n$.", " {\\it The directional derivative of $f$ at $\\mathbf{X}$ in the", "direction of\\/} $\\boldsymbol{\\Phi}$ is defined by", "$$", "\\frac{\\partial f(\\mathbf{X})}{\\partial\\boldsymbol{\\Phi}}=\\lim_{t\\to", "0}\\frac", "{f(\\mathbf{X}+ t\\boldsymbol{\\Phi})-f(\\mathbf{X})}{ t}", "$$", "if the limit exists. That is, $\\partial f(\\mathbf{X})/\\partial\\boldsymbol{\\Phi}$", "is the ordinary derivative of the function", "$$", "h(t)=f(\\mathbf{X}+t\\boldsymbol{\\Phi})", "$$", "at $t=0$, if $h'(0)$ exists." ], "refs": [], "ref_ids": [] }, { "id": 350, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.3.5", "categories": [], "title": "", "contents": [ "A function $f$ is {\\it differentiable\\/} at", "$$", " \\mathbf{X}_0=(x_{10},x_{20}, \\dots,x_{n0}))", "$$", "if $\\mathbf{X}_0\\in D_f^0$ and", "there are constants $m_1$, $m_2$, \\dots$,$ $m_n$ such that", "\\begin{equation}\\label{eq:5.3.16}", "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} \\frac{f(\\mathbf{X})-f(\\mathbf{X}_0)-", "\\dst{\\sum^n_{i=1}}\\, m_i (x_i-x_{i0})}{ |\\mathbf{X}-\\mathbf{X}_0|}=0.", "\\end{equation}" ], "refs": [], "ref_ids": [] }, { "id": 351, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.1", "categories": [], "title": "", "contents": [ "A vector-valued function", " $\\mathbf{G}=(g_1,g_2, \\dots,g_n)$ is {\\it", "differentiable\\/} at", "$$", "\\mathbf{U}_0=(u_{10},u_{20}, \\dots,u_{m0})", "$$", " if its component functions", "$g_1$, $g_2$, \\dots, $g_n$ are differentiable at $\\mathbf{U}_0$.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 352, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.7", "categories": [], "title": "", "contents": [ "Suppose that $r\\ge1$ and all partial derivatives of $f$ of order $\\le r-1$", "are differentiable in a neighborhood of $\\mathbf{X}_0$. Then the $r$th", "{\\it differential of $f$ at\\/} $\\mathbf{X}_0$, denoted by $d^{(r)}_{\\mathbf{X}_0}f$, is defined by", "\\begin{equation} \\label{eq:5.4.23}", "d^{(r)}_{\\mathbf{X}_0}f=\\sum_{i_1,i_2, \\dots,i_r=1}^n", "\\frac{\\partial^rf(\\mathbf{X}_0)", "}{\\partial x_{i_r}\\partial x_{i_{r-1}}\\cdots\\partial x_{i_1}}", "dx_{i_1}dx_{i_2}\\cdots dx_{i_r},", "\\end{equation}", "where $dx_1$, $dx_2$, \\dots, $dx_n$ are the differentials", "introduced in Section~5.3; that is, $dx_i$ is the function", "whose value at a point in $\\R^n$ is the $i$th coordinate", "of the point.", "For convenience, we define", "$$", "(d^{(0)}_{\\mathbf{X}_0}f)=f(\\mathbf{X}_0).", "$$", "Notice that $d^{(1)}_{\\mathbf{X}_0}f=d_{\\mathbf{X}_0}f$.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 353, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.1", "categories": [], "title": "", "contents": [ "A transformation $\\mathbf{L}: \\R^n \\to \\R^m$", "defined on all of", "$\\R^n$ is {\\it linear\\/} if", "$$", "\\mathbf{L}(\\mathbf{X}+\\mathbf{Y})=\\mathbf{L}(\\mathbf{X})+\\mathbf{L}(\\mathbf{Y})", "$$", "for all $\\mathbf{X}$ and $\\mathbf{Y}$ in $\\R^n$ and", "$$", "\\mathbf{L}(a\\mathbf{X})=a\\mathbf{L}(\\mathbf{X})", "$$", "for all $\\mathbf{X}$ in $\\R^n$ and real numbers $a$." ], "refs": [], "ref_ids": [] }, { "id": 354, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.3", "categories": [], "title": "", "contents": [ "\\begin{alist}", "\\item % (a)", " If $c$ is a real number and", "$\\mathbf{A}=[a_{ij}]$ is an $m\\times n$ matrix, then $c\\mathbf{A}$ is the", "$m\\times n$ matrix defined by", "$$", "c\\mathbf{A}=[ca_{ij}];", "$$", "that is, $c\\mathbf{A}$ is obtained by multiplying every entry of", "$\\mathbf{A}$ by $c$.", "\\item % (b)", "If $\\mathbf{A}=[a_{ij}]$ and $\\mathbf{B}=[b_{ij}]$ are $m\\times n$", "matrices, then the {\\it sum\\/}", " $\\mathbf{A}+ \\mathbf{B}$", " is the", "$m\\times n$ matrix", "$$", "\\mathbf{A}+\\mathbf{B}=[a_{ij}+b_{ij}];", "$$", "that is, the sum of two $m\\times n$ matrices is obtained by adding", "corresponding entries. The sum of two matrices is not defined unless", "they have the same number of rows and the same number of columns.", "\\item % (c)", "If $\\mathbf{A}=[a_{ij}]$ is an $m\\times p$ matrix and $\\mathbf{B}= [b_{ij}]$", "is a $p\\times n$ matrix, then the {\\it product\\/}", "$\\mathbf{C}=\\mathbf{AB}$ is the $m\\times n$ matrix with", "$$", "c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\\cdots+a_{ip}b_{pj}=\\sum^p_{k=1}", "a_{ik}b_{kj},\\quad 1\\le i\\le m,\\ 1\\le j\\le n.", "$$", "Thus, the $(i,j)$th entry of $\\mathbf{AB}$ is obtained by", "multiplying each entry in the $i$th row of $\\mathbf{A}$ by the", "corresponding entry in the $j$th column of $\\mathbf{B}$ and adding the", "products. This definition requires that $\\mathbf{A}$ have the same number", "of columns as $\\mathbf{B}$ has rows. Otherwise, $\\mathbf{AB}$ is", "undefined.", "\\end{alist}" ], "refs": [], "ref_ids": [] }, { "id": 355, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.8", "categories": [], "title": "", "contents": [ "The {\\it norm\\/}$,$ $\\|\\mathbf{A}\\|,$ of an $m\\times n$ matrix", "$\\mathbf{A}=[a_{ij}]$ is the smallest number such that", "$$", "|\\mathbf{AX}|\\le\\|\\mathbf{A}\\|\\,|\\mathbf{X}|", "$$", "for all $\\mathbf{X}$ in $\\R^n.$", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 356, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.10", "categories": [], "title": "", "contents": [ "Let $\\mathbf{A}=[a_{ij}]$ be an $n\\times n$ matrix$,$ with $n\\ge2.$", "The {\\it cofactor\\/} of an entry $a_{ij}$ is", "$$", "c_{ij}=(-1)^{i+j}\\det(\\mathbf{A}_{ij}),", "$$", "where $\\mathbf{A}_{ij}$ is the $(n-1)\\times(n-1)$ matrix obtained by", "deleting the $i$th row and $j$th column of $\\mathbf{A}.$", "The {\\it adjoint\\/} of", "$\\mathbf{A},$ denoted by", "$\\adj(\\mathbf{A}),$ is the", "$n\\times n$ matrix whose $(i,j)$th entry is $c_{ji}.$" ], "refs": [], "ref_ids": [] }, { "id": 357, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:6.3.2", "categories": [], "title": "", "contents": [ "A transformation $\\mathbf{F}: \\R^n\\to \\R^n$ is", "{\\it regular\\/} on an open set $S$ if $\\mathbf{F}$ is one-to-one and", "continuously", "differentiable on $S$, and $J\\mathbf{F}(\\mathbf{X})\\ne0$ if $\\mathbf{X}\\in S$.", "We will also say that $\\mathbf{F}$", " is regular on an arbitrary set $S$ if", "$\\mathbf{F}$ is regular on an open set containing $S$." ], "refs": [], "ref_ids": [] }, { "id": 358, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.1", "categories": [], "title": "", "contents": [ "A {\\it coordinate rectangle\\/} $R$ in $\\R^n$ is the Cartesian", "product of $n$ closed intervals; that is,", "$$", "R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n].", "$$", "The {\\it content\\/} of $R$ is", "$$", "V(R)=(b_1-a_1)(b_2-a_2)\\cdots (b_n-a_n).", "$$", "The numbers $b_1-a_1$, $b_2-a_2$, \\dots, $b_n-a_n$ are the {\\it edge", "lengths\\/} of $R$. If", "they are equal, then", "$R$ is a", "{\\it coordinate cube\\/}.", " If $a_r=b_r$ for some $r$, then $V(R)=0$ and we", "say that $R$ is {\\it degenerate\\/};", "otherwise,", "$R$ is", "{\\it nondegenerate\\/}.", " \\bbox" ], "refs": [], "ref_ids": [] }, { "id": 359, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.2", "categories": [], "title": "", "contents": [ "Let $f$ be a real-valued function defined", "on a rectangle $R$ in $\\R^n$. We say that", " $f$ is {\\it Riemann integrable on\\/} $R$", " if there is a number $L$ with the following property: For", "every $\\epsilon>0$, there is a $\\delta>0$ such that", "$$", "\\left|\\sigma-L\\right|<\\epsilon", "$$", "if $\\sigma$ is any Riemann sum of $f$ over", "a partition ${\\bf P}$ of $R$", "such that $\\|{\\bf P}\\|<\\delta$.", "In this case, we say that", " $L$ is the {\\it Riemann integral of $f$ over\\/} $R$, and write", "$$", "\\int_R f(\\mathbf{X})\\,d\\mathbf{X}=L.", "\\eqno{\\bbox}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 360, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.4", "categories": [], "title": "", "contents": [ "If $f$ is bounded on a rectangle $R$ in $\\R^n$ and", "${\\bf P}=\\{R_1,R_2, \\dots,R_k\\}$ is a partition of $R$, let", "$$", "M_j=\\sup_{\\mathbf{X}\\in R_j}f(\\mathbf{X}),\\quad m_j=", "\\inf_{\\mathbf{X}\\in R_j}f(\\mathbf{X}).", "$$", "The {\\it upper sum\\/} of $f$ over ${\\bf P}$ is", "$$", "S({\\bf P})=\\sum_{j=1}^k M_jV(R_j),", "$$", "and the {\\it upper integral", " of $f$ over\\/} $R$, denoted by", "$$", "\\overline{\\int_R}\\,f(\\mathbf{X})\\,d\\mathbf{X},", "$$", " is the infimum of all upper", "sums. The {\\it lower sum of $f$ over\\/} ${\\bf P}$ is", "$$", "s({\\bf P})=\\sum_{j=1}^k m_jV(R_j),", "$$", "and the {\\it lower integral", " of $f$ over \\/}$R$, denoted by", "$$", "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X},", "$$", " is the supremum of all lower sums.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 361, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.14", "categories": [], "title": "", "contents": [ "A subset $E$ of $\\R^n$ has zero content if for each", "$\\epsilon>0$", "there is a finite set of rectangles $T_1$, $T_2$, \\dots, $T_m$ such", "that", "\\begin{equation}\\label{eq:7.1.24}", "E\\subset\\bigcup_{j=1}^m T_j", "\\end{equation}", "and", "\\begin{equation}\\label{eq:7.1.25}", "\\sum_{j=1}^m V(T_j)<\\epsilon.", "\\end{equation}" ], "refs": [], "ref_ids": [] }, { "id": 362, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.17", "categories": [], "title": "", "contents": [ "Suppose that $f$ is bounded on a bounded subset of $S$ of", "$\\R^n$, and let", "\\begin{equation}\\label{eq:7.1.36}", "f_S(\\mathbf{X})=\\left\\{\\casespace\\begin{array}{ll} f(\\mathbf{X}),&\\mathbf{X}\\in", "S,\\\\[2\\jot]", " 0,&\\mathbf{X}\\not\\in S.\\end{array}\\right.", "\\end{equation}", "Let $R$ be a rectangle containing $S$.", "Then {\\it the integral of $f$ over $S$\\/} is defined to be", "$$", "\\int_S f(\\mathbf{X})\\,d\\mathbf{X}=\\int_R f_S(\\mathbf{X})\\,d\\mathbf{X}", "$$", "if $\\int_R f_S(\\mathbf{X})\\,", "d\\mathbf{X}$ exists.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 363, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.18", "categories": [], "title": "", "contents": [ "If $S$ is a bounded subset of $\\R^n$ and", "the integral $\\int_S\\,d\\mathbf{X}$ (with integrand $f\\equiv1$)", "exists, we call $\\int_S\\,d\\mathbf{X}$ the {\\it content\\/} (also, {\\it area\\/} if", "$n=2$ or", "{\\it volume\\/} if $n=3$) of $S$, and denote it by $V(S)$;", "thus,", "$$", "V(S)=\\int_S\\,d\\mathbf{X}.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 364, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.20", "categories": [], "title": "", "contents": [ "A {\\it differentiable surface\\/} $S$ in $\\R^n\\ (n>1)$ is the", "image of a", "compact subset $D$ of $\\R^m$, where $m< n$, under a continuously", "differentiable transformation $\\mathbf{G}: \\R^m\\to \\R^n$. If", "$m=1$, $S$ is also called a {\\it differentiable curve\\/}." ], "refs": [], "ref_ids": [] }, { "id": 365, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.9", "categories": [], "title": "", "contents": [ "If $\\mathbf{A}=[a_{ij}]$ is an $n \\times n$ matrix$,$ then", "$$", "\\max\\set{\\sum_{j=1}^n |a_{ij}|}{1\\le i\\le n}", "$$", "is the {\\it infinity norm of\\/} $A,$ denoted by $\\|A\\|_\\infty$." ], "refs": [], "ref_ids": [] }, { "id": 366, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.1", "categories": [], "title": "", "contents": [ "A {\\it metric space\\/} is a nonempty set $A$ together with", "a real-valued function $\\rho$ defined on $A\\times A$ such that", " if $u$, $v$, and $w$", "are arbitrary members of $A$, then", "\\begin{alist}", "\\item % (a)", "$\\rho(u,v)\\ge 0$, with equality if and only if $u=v$;", "\\item % (b)", "$\\rho(u,v)=\\rho(v,u)$;", "\\item % (c)", "$\\rho(u,v)\\le\\rho(u,w)+\\rho(w,v)$.", "\\end{alist}", "We say that $\\rho$ is a {\\it metric\\/} on $A$.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 367, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.2", "categories": [], "title": "", "contents": [ "A {\\it vector space\\/} $A$", "is a nonempty set of elements called", "{\\it vectors\\/} on which two operations, vector", "addition and scalar multiplication", "(multiplication by real numbers) are defined, such", "that the following assertions are true for all $\\mathbf{U}$, $\\mathbf{V}$,", "and $\\mathbf{W}$ in $A$ and all real numbers $r$ and $s$:\\\\", "\\phantom{1}1. $\\mathbf{U}+\\mathbf{V}\\in A$;\\\\", "\\phantom{1}2. $\\mathbf{U}+\\mathbf{V}=\\mathbf{V}+\\mathbf{U}$;\\\\", "\\phantom{1}3. $\\mathbf{U}+(\\mathbf{V}+\\mathbf{W})=(\\mathbf{U}+\\mathbf{V})+\\mathbf{W}$;\\\\", "\\phantom{1}4. There is a vector $\\mathbf{0}$ in $A$", "such that $\\mathbf{U}+\\mathbf{0}=\\mathbf{U}$;\\\\", "\\phantom{1}5. There is a vector $-\\mathbf{U}$ in $A$", "such that $\\mathbf{U}+(-\\mathbf{U})=\\mathbf{0}$;\\\\", "\\phantom{1}6. $r\\mathbf{U}\\in A$;\\\\", "\\phantom{1}7. $r(\\mathbf{U}+\\mathbf{V})=r\\mathbf{U}+r\\mathbf{V}$;\\\\", "\\phantom{1}8. $(r+s)\\mathbf{U}=r\\mathbf{U}+s\\mathbf{U}$;\\\\", "\\phantom{1}9. $r(s\\mathbf{U})=(rs)\\mathbf{U}$; \\\\", "10. $1\\mathbf{U}=\\mathbf{U}$.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 368, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.3", "categories": [], "title": "", "contents": [ "A {\\it normed vector space\\/}", " is a vector space", "$A$ together with a real-valued function $N$ defined on", "$A$, such that", " if $u$ and $v$", "are arbitrary vectors in $A$ and $a$ is a real number, then", "\\begin{alist}", "\\item % (a)", "$N(u)\\ge 0$ with equality if and only if $u=0$;", "\\item % (b)", "$N(au)=|a|N(u)$;", "\\item % (c)", "$N(u+v)\\le N(u)+N(v)$.", "\\end{alist}", "We say that $N$ is a {\\it norm\\/} on", "$A$, and", "$(A,N)$ is a {\\it normed vector space\\/}." ], "refs": [], "ref_ids": [] }, { "id": 369, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.6", "categories": [], "title": "", "contents": [ "If $p\\ge 1$ and $\\mathbf{X}=(x_1,x_2, \\dots,x_n)$, let", "\\begin{equation} \\label{eq:8.1.3}", "\\|\\mathbf{X}\\|_p", "=\\left(\\sum_{i=1}^n|x_i|^p\\right)^{1/p}.", "\\end{equation}", "The metric induced on $\\R^n$ by this norm is", "$$", "\\rho_p(\\mathbf{X},\\mathbf{Y})", "=\\left(\\sum_{i=1}^n|x_i-y_i|^p\\right)^{1/p}.", "\\eqno{\\bbox}", "$$" ], "refs": [], "ref_ids": [] }, { "id": 370, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.10", "categories": [], "title": "", "contents": [ "If $u_0\\in A$ and $\\epsilon>0$, the set", "$$", "N_\\epsilon(u_0)=\\set{u\\in A}{\\rho(u_0,u)<\\epsilon}", "$$", "is called an {\\it $\\epsilon$-neighborhood\\/} of $u_0$.", "(Sometimes we call $S_\\epsilon$ the {\\it open ball of radius", "$\\epsilon$ centered at $u_0$\\/}.)", "If a subset $S$ of $A$ contains an $\\epsilon$-neighborhood of $u_0$,", "then", "$S$ is a {\\it neighborhood\\/} of", "$u_0$, and", "$u_0$ is an", "{\\it interior point\\/} of", "$S$. The set of interior points of", "$S$ is the {\\it interior\\/} of $S$,", "denoted by", "$S^0$. If every", "point of $S$ is an interior point", "(that is,", "$S^0=S$), then", "$S$ is", "{\\it open\\/}. A set $S$ is {\\it closed\\/} if", "$S^c$ is open." ], "refs": [], "ref_ids": [] }, { "id": 371, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.12", "categories": [], "title": "", "contents": [ " Then", "\\begin{alist}", "\\item % (a)", "$u_0$ is a {\\it limit point\\/} of $S$ if every deleted neighborhood of", "$u_0$ contains a point of~$S$.", "\\item % (b)", "$u_0$ is a {\\it boundary", "point\\/} of $S$ if every neighborhood of $u_0$ contains at least one point", "in $S$ and one not in $S$. The set of boundary points of $S$ is the {\\it", "boundary\\/} of $S$, denoted by $\\partial S$. The {\\it closure\\/} of $S$,", "denoted by $\\overline{S}$, is defined by $\\overline{S}=S\\cup \\partial S$.", "\\item % (c)", "$u_0$ is an {\\it isolated point\\/} of $S$ if $u_0\\in S$ and there is a", "neighborhood of $u_0$ that contains no other point of $S$.", "\\item % (d)", "$u_0$ is {\\it exterior } to $S$ if $u_0$ is in the interior of $S^c$. The", "collection of such points is the {\\it exterior\\/} of $S$.", "\\bbox", "\\end{alist}" ], "refs": [], "ref_ids": [] }, { "id": 372, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.14", "categories": [], "title": "", "contents": [ "A sequence $\\{u_n\\}$ in a metric space $(A,\\rho)$", " {\\it converges\\/}", "to", "$u\\in A$ if", "\\begin{equation} \\label{eq:8.1.16}", "\\lim_{n\\to\\infty}\\rho(u_n,u)=0.", "\\end{equation}", "In this case we say that", "$\\lim_{n\\to\\infty}u_n=u$.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 373, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.16", "categories": [], "title": "", "contents": [ "A sequence $\\{u_n\\}$ in a metric space $(A,\\rho)$ is", " a {\\it Cauchy sequence\\/}", " if for every", "$\\epsilon>0$ there is an integer $N$ such that", "\\begin{equation} \\label{eq:8.1.17}", "\\rho(u_n,u_m)<\\epsilon\\mbox{\\quad and \\quad}m,n>N.", "\\end{equation}" ], "refs": [], "ref_ids": [] }, { "id": 374, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.18", "categories": [], "title": "", "contents": [ "A metric space $(A,\\rho)$ is {\\it complete\\/}", " if every Cauchy sequence in $A$", "has a limit." ], "refs": [], "ref_ids": [] }, { "id": 375, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.20", "categories": [], "title": "", "contents": [ "If $\\rho$ and $\\sigma$ are both metrics on a set $A$, then $\\rho$", "and $\\sigma$ are {\\it equivalent \\/}", "\\hskip-.2em if there are positive constants $\\alpha$ and $\\beta$", "such that", "\\begin{equation} \\label{eq:8.1.18}", "\\alpha\\le\\frac{\\rho(x,y)}{\\sigma(x,y)}\\le\\beta", "\\mbox{\\quad for all \\quad}x,y\\in A\\mbox{\\quad such that \\quad}x\\ne y.", "\\end{equation}" ], "refs": [], "ref_ids": [] }, { "id": 376, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.1", "categories": [], "title": "", "contents": [ "The {\\it diameter\\/} of a nonempty subset $S$ of $A$ is", "$$", "d(S)=\\sup\\set{\\rho(u,v)}{u,\\, v\\in T}.", "$$", "If $d(S)<\\infty$ then $S$ is {\\it bounded\\/}.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 377, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.2", "categories": [], "title": "", "contents": [ "A set $T$ is {\\it compact\\/} if", "it has the Heine--Borel property." ], "refs": [], "ref_ids": [] }, { "id": 378, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.7", "categories": [], "title": "", "contents": [ "A set $T$ is {\\it totally bounded\\/}", " if for every", "$\\epsilon>0$", "there is a finite set $T_\\epsilon$ with the following property:", "if $t\\in T$, there is an $s\\in T_\\epsilon$ such that", "$\\rho(s,t)<\\epsilon$.", "We say that $T_\\epsilon$ is a {\\it finite $\\epsilon$-net for $T$\\/}.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 379, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.10", "categories": [], "title": "", "contents": [ "A subset $T$ of $C[a,b]$ is {\\it uniformly bounded\\/} if there is a", "constant $M$ such that", "\\begin{equation} \\label{eq:8.2.6}", "|f(x)|\\le M \\mbox{\\quad if \\quad} a\\le x\\le b\\mbox{\\quad and \\quad}", "f\\in T.", "\\end{equation}", "A subset $T$ of $C[a,b]$ is {\\it", "equicontinuous\\/} if for each", "$\\epsilon>0$ there is a $\\delta>0$ such that", "\\begin{equation} \\label{eq:8.2.7}", "|f(x_1)-f(x_2)|\\le \\epsilon \\mbox{\\quad if \\quad}", "x_1,x_2\\in [a,b],\\quad |x_1-x_2|<\\delta,\\mbox{\\quad and \\quad}f\\in T.", "\\end{equation}" ], "refs": [], "ref_ids": [] }, { "id": 380, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.1", "categories": [], "title": "", "contents": [ "We say that", "$$", "\\lim_{u\\to \\widehat u}f(u)=\\widehat v", "$$", "if $\\widehat u\\in\\overline D_f$ and for each $\\epsilon>0$ there is a", "$\\delta>0$ such that", "\\begin{equation} \\label{eq:8.3.1}", "\\sigma(f(u),\\widehat v)<\\epsilon\\mbox{\\quad if \\quad}", "u\\in D_f", "\\mbox{\\quad and \\quad}", "0<\\rho(u,\\widehat u)<\\delta.", "\\end{equation}" ], "refs": [], "ref_ids": [] }, { "id": 381, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.2", "categories": [], "title": "", "contents": [ "We say that $f$", "is {\\it continuous\\/} at", "$\\widehat u$ if", "$\\widehat u\\in D_f$ and for each $\\epsilon>0$", "there is a $\\delta>0$ such that", "\\begin{equation} \\label{eq:8.3.2}", "\\sigma(f(u),f(\\widehat u))<\\epsilon\\mbox{\\quad if \\quad}", "u\\in D_f\\cap N_\\delta(\\widehat u).", "\\end{equation}", "If $f$ is continuous at every point of a set $S$,", "then $f$ is {\\it continuous on\\/} S.", "\\bbox" ], "refs": [], "ref_ids": [] }, { "id": 382, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.7", "categories": [], "title": "", "contents": [ "A function $f$ is {\\it uniformly continuous\\/} on a subset $S$ of $D_f$ if", "for each $\\epsilon>0$ there is a $\\delta>0$ such that", "$$", "\\sigma(f(u),f(v))<\\epsilon\\mbox{\\quad whenever \\quad}", "\\rho(u,v)<\\delta\\mbox{\\quad and \\quad}u,v\\in S.", "$$" ], "refs": [], "ref_ids": [] }, { "id": 383, "type": "definition", "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.9", "categories": [], "title": "", "contents": [ "If $f:(A,\\rho)\\to (A,\\rho)$ is defined on all of $A$", "and there is a constant $\\alpha$ in $(0,1)$", "such that", "\\begin{equation} \\label{eq:8.3.7}", "\\rho(f(u),f(v))\\le\\alpha\\rho(u,v)", "\\mbox{\\quad for all\\quad} (u,v)\\in A\\times A,", "\\end{equation}", "then $f$ is a {\\it contraction\\/} of $(A,\\rho)$.", "\\bbox" ], "refs": [], "ref_ids": [] } ], "others": [], "retrieval_examples": [ 3, 4, 7, 8, 11, 12, 13, 14, 17, 22, 26, 27, 29, 30, 31, 32, 33, 35, 39, 43, 44, 45, 47, 48, 50, 51, 52, 53, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 80, 81, 83, 85, 86, 87, 88, 89, 91, 93, 95, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 112, 113, 114, 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