Question 80
Let $A = (a_{1}, a_{2}, \ldots, a_{n})$ be a sequence of integers. The successor of $A$ is the sequence $A' = (a_{1}', a_{2}', \ldots, a_{n}')$ with
$$
a_{k}' = \left|\{i \lt k \mid a_{i} \lt a_{k}\}\right| - \left|\{i \gt k \mid a_{i} \gt a_{k}\}\right|
$$
Let $A_{0}$ be a finite sequence of integers and for $k \geq 0$ let $A_{k+1} = A_{k}'$ be the successor of $A_{k}$. Show that there exists a natural number $m$ with $A_{m} = A_{m+1}$.
$$
a_{k}' = \left|\{i \lt k \mid a_{i} \lt a_{k}\}\right| - \left|\{i \gt k \mid a_{i} \gt a_{k}\}\right|
$$
Let $A_{0}$ be a finite sequence of integers and for $k \geq 0$ let $A_{k+1} = A_{k}'$ be the successor of $A_{k}$. Show that there exists a natural number $m$ with $A_{m} = A_{m+1}$.