2012 IMO Shortlist C1

Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x, y)$ by either $(y+1, x)$ or $(x-1, x)$. Prove that she can perform only finitely many such iterations.

几个正整数连续写入。迭代地，Alice 选择两个相邻的数字 $x$ 和 $y$，使得 $x>y$ 和 $x$ 位于 $y$ 的左侧，并用 $(y+1, x)$ 或 $(x-1, x)$ 替换 $(x, y)$ 对。证明她只能执行有限多次这样的迭代。