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At least one of two limits is equal to infinity

Source: ICMC 7 Round 1 Problem 6

January 8, 2024
number theorylimitsbijectionICMC

Problem Statement

Let f:N→Nf:\mathbb{N}\to\mathbb{N} be a bijection of the positive integers. Prove that at least one of the following limits is true: lim⁡N→∞∑n=1N1n+f(n)=∞;lim⁡N→∞∑n=1N(1n−1f(n))=∞.\lim_{N\to\infty}\sum_{n=1}^{N}\frac{1}{n+f(n)}=\infty;\qquad\lim_{N\to\infty}\sum_{n=1}^N\left(\frac{1}{n}-\frac{1}{f(n)}\right)=\infty.Proposed by Dylan Toh
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