Question 33
Let $z_{0} \lt z_{1} \lt z_{2} \lt \cdots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n \geqslant 1$ such that
$$
z_{n} \lt \frac{z_{0} + z_{1} + \cdots + z_{n}}{n} \leqslant z_{n+1} .
$$
$$
z_{n} \lt \frac{z_{0} + z_{1} + \cdots + z_{n}}{n} \leqslant z_{n+1} .
$$