Question 150
Let $N$ be a positive integer and let $a_{1}, a_{2}, \ldots$ be an infinite sequence of positive integers. Suppose that, for each $n \gt N$, $a_{n}$ is equal to the number of times $a_{n-1}$ appears in the list $a_{1}, a_{2}, \ldots, a_{n-1}$.
Prove that at least one of the sequences $a_{1}, a_{3}, a_{5}, \ldots$ and $a_{2}, a_{4}, a_{6}, \ldots$ is eventually periodic.
Prove that at least one of the sequences $a_{1}, a_{3}, a_{5}, \ldots$ and $a_{2}, a_{4}, a_{6}, \ldots$ is eventually periodic.