Let $p$ be a permutation of $\mathbb{N}$. We say that $p$ is bounded if there exists $k$ so that $|p(i)-i|\le k$ for all $i$. If $p$ is bounded, must there exist $M>0$ such that $p(\{1,2,\dots ,M\})=\{1,2,\dots ,M\}$? If so, can we bound $M$ by a function of $k$?