Question 1
Let $a_0 \lt a_1 \lt a_2 \lt \dotsb$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that $$
a_n \lt \frac{a_0+a_1+a_2+\dotsb+a_n}{n} \le a_{n+1}.
$$
a_n \lt \frac{a_0+a_1+a_2+\dotsb+a_n}{n} \le a_{n+1}.
$$