USAMO 1995 Problem 4

$a_{0}, a_{1}, a_{2}, \ldots$ is an infinite sequence of integers such that $a_{n}-a_{m}$ is divisible by $n-m$ for all (unequal) $n$ and $m$. For some polynomial $\mathrm{p}(\mathrm{x})$ we have $\mathrm{p}(\mathrm{n})>\left|\mathrm{a}_{\mathrm{n}}\right|$ for all $\mathrm{n}$. Show that there is a polynomial $\mathrm{q}(\mathrm{x})$ such that $\mathrm{q}(\mathrm{n})=$ $a_{n}$ for all $n$.