MathLabs

Problem 1

Let a0<a1<a2<⋯a_0<a_1<a_2<\cdots be an infinite sequence of positive integers. Prove that there exists a unique integer n≥1n\ge1 such that an<a0+a1+⋯+ann≤an+1.a_n<\frac{a_0+a_1+\cdots+a_n}{n}\le a_{n+1}.
Step 5 of 5: Conclusion: a unique sign change
Previous stepNext step
∃! n≥1:dn>0≥dn+1\exists! \, n\ge1 : d_n > 0 \ge d_{n+1}
Detailed analysis

A strictly decreasing sequence of integers starting at a positive value d1>0d_1>0 must eventually become non-positive, and since it strictly decreases it can cross from positive to non-positive at only one index. That unique index is exactly the last one with dn>0d_n>0, so dn>0≥dn+1d_n>0\ge d_{n+1} holds for exactly one n≥1n\ge1, proving both existence and uniqueness of the required nn.