Problem 1

Let

a_0<a_1<a_2<\cdots

be an infinite sequence of positive integers. Prove that there exists a unique integer

n\ge1

such that

a_n<\frac{a_0+a_1+\cdots+a_n}{n}\le a_{n+1}.

Step 1 of 5: Introduce the difference d_n

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In plain words

Multiplying the target inequalities by $n$ turns them into statements about integers, which is easier to control than the fraction directly.

d_n := (a_0+a_1+\cdots+a_n) - n a_n

Detailed analysis

For $n\ge1$ define $d_n=(a_0+a_1+\cdots+a_n)-na_n$ , an integer since all $a_i$ are integers. The inequality $a_n<\frac{a_0+\cdots+a_n}{n}$ is equivalent to $na_n<a_0+\cdots+a_n$ , i.e. $d_n>0$ .