Question 83
Let $f:\mathbb N\to\mathbb N$, and let $f^m$ denote $m$ iterations of $f$. Suppose that for every $n\in\mathbb N$ there exists a $k$ such that $f^{2k}(n)=n+k$, and let $k_n$ be the smallest such $k$. Prove that the sequence $k_1,k_2,\dots$ is unbounded.