On the impossibility of an infinite chain in The Collatz Conjecture

The Collatz Conjecture defines a number sequence constructed as follows:

If X_1 is odd, then X_2 = 3X_1 + 1.

If X_1 is even, then X_2 = X_1 / 2.

The conjecture itself asserts that every such sequence, starting from any number, eventually reaches the number 1 (or, more precisely, the cycle 4-2-1).

There are two hypothetical exceptions - either another cycle other than 4-2-1, or an infinite divergent sequence.

In this paper, we will prove that the second option is impossible.