Question 15
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} $, ... and $ a_{2} $, $ a_{4} $, $ a_{6} $, ... is eventually periodic.
Prove that at least one of the sequences $ a_{1} $, $ a_{3} $, $ a_{5} $, ... and $ a_{2} $, $ a_{4} $, $ a_{6} $, ... is eventually periodic.