Integrality and boundedness implies periodicity

Question

Let's consider the recurrence $a_{n+r}=c_{1}a_{n+r-1}+\ldots+c_{r}a_{n}, c_{i}\in \mathbb{Z}$. How to prove that if the $a_{n}$ is integer and bounded, then it's periodic?

Could someone suggest the possible approach to the problem?

Answer 1

Consider the set of all $r$-tuples $(a_n, a_{n+1}, \ldots, a_{n+r-1})$ and use the pigeonhole principle.