MathNet 2014 Question 62

A doubly infinite sequence $a_n$, for $n \in \mathbb{Z}$, has each $a_n$ equal to either $0$ or $1$. Prove that there exist numbers $p$ and $q \gt 1$ such that $a_{p+k} = a_{p+q+k}$ for $k = 0, 1, \dots, q-1$.