Question 160
Let $a_{0} \lt a_{1} \lt a_{2} \lt \ldots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n \geqslant 1$ such that
$$
a_{n} \lt \frac{a_{0} + a_{1} + a_{2} + \cdots + a_{n}}{n} \leqslant a_{n+1}.
$$
$$
a_{n} \lt \frac{a_{0} + a_{1} + a_{2} + \cdots + a_{n}}{n} \leqslant a_{n+1}.
$$