Question 1
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
$$
z_{n}\lt \frac{z_{0}+z_{1}+\cdots+z_{n}}{n} \leqslant z_{n+1} .
$$
$$
z_{n}\lt \frac{z_{0}+z_{1}+\cdots+z_{n}}{n} \leqslant z_{n+1} .
$$