International Mathematical Olympiad 2014 Problem 1

Let $a_0 < a_1 < a_2 < \cdots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n \geq 1$ such that $$a_n < \frac{a_0 + a_1 + \cdots + a_n}{n} \leq a_{n+1}.$$