IMO 2014 Problem 1

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

Proposed by Gerhard Wöginger, Austria.