Let
![a_0 < a_1 < a_2 < \cdots](/media/m/6/c/f/6cff7853117a6eed7b491223d5385829.png)
be an infinite sequence of positive integers. Prove that there exists a unique integer
![n \geq 1](/media/m/a/9/8/a982fcac3e2c9e0d94e965d6efb5a582.png)
such that
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Let $a_0 < a_1 < a_2 < \cdots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n \geq 1$ such that $$
a_n < \frac{a_0 + a_1 + \cdots + a_n}{n} \leq a_{n+1} \text{.}
$$