Let $n$ be a positive integer. Suppose $S$ is a set of ordered $n-\mbox{tuples}$ of nonnegative integers such that, whenever $(a_1,\dots,an)\in S$ and $b_i$ are nonnegative integers with$b_i\le a_i$, the $n-\text{tuple}$ $(b_1,\dots,b_n)$ is also in $S$. If $h_m$ is the number of elements of $S$ with the sum of components equal to$m$, prove that $h_m$ is a polynomial in $m$ for all sufficiently large$m$.