{ "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", "$$", " for all integers $n$. Hence,", " $\\beta-\\epsilon$ is an upper bound of $S$. Since $\\beta-\\epsilon", "<\\beta$, this contradicts the definition of~$\\beta$." ], "refs": [], "ref_ids": [] } ], "ref_ids": [] }, { "id": 3, "type": "theorem", "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.6", "categories": [], "title": "", "contents": [ "The rational numbers are dense in the reals$\\,;$ that is, if $a$", "and", "$b$ are real numbers with $a1$. There is also an integer", "$j$ such that $j>qa$. This is obvious if $a\\le0$, and it follows from", "Theorem~\\ref{thmtype:1.1.4} with $\\epsilon=1$ and $\\rho=qa$ if $a>0$. Let", "$p$ be the smallest integer such that $p>qa$. Then $p-1\\le qa$, so", "$$", "qa
0$ $($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",
"$$",
"\\alpha=\\inf_{a0$ there is",
"a partition $P$ of $[a,b]$ for which",
"\\begin{equation} \\label{eq:3.2.19}",
"S(P)-s(P)<\\epsilon.",
"\\end{equation}"
],
"refs": [],
"proofs": [
{
"contents": [
"We leave it to you (Exercise~\\ref{exer:3.2.4}) to show that if $\\int_a^b",
"f(x)\\,dx$ exists, then \\eqref{eq:3.2.19} holds for $\\|P\\|$ sufficiently",
"small. This implies that the stated condition is necessary for",
"integrability. To show that it is sufficient, we observe that since",
"$$",
"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}).",
"$$",
"Since $0\\left(1-\\frac{1}{ n}\\right)^q,\\quad n\\ge k,",
"$$",
"\\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 $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 $r