ISL 2014 A1

Let $z_{0}<z_{1}<z_{2}<\cdots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n \geqslant 1$ such that $$z_{n}<\frac{z_{0}+z_{1}+\cdots+z_{n}}{n} \leqslant z_{n+1}$$ (Austria)