MathLabs

Problem 5

Let ff be an injective function from {1,2,3,…}\{1,2,3,\ldots\} into itself. Prove that for any nn we have ∑k=1nf(k)k−2≥∑k=1nk−1\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.
Step 1 of 4: Restrict to the first n values
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In plain words

The problem is a weighted sum of n distinct positive integers.

ak=f(k)(1≤k≤n)a_k=f(k)\quad(1\le k\le n)
Detailed analysis

The numbers a1,…,ana_1,\ldots,a_n are distinct positive integers because ff is injective.