## A Cute Solution to a Cute Problem

*Date Written: August 30, 2023; Last Modified: August 30, 2023*

I previously posted about a simple problem that didn’t seem to have a very simple solution:

**Problem 1. **

Let \(f\) be a function satisfying \(0\leq f(x)\leq f(y)\) if \(x\geq y\). Suppose \(\int_0^\infty f(x) dx\) converges. Prove \(\lim_{x\to\infty} xf(x)=0\).

I came up with a solution that relied on the dominated convergence theorem, but that wouldn’t really be acceptable for most undergraduate-level analysis courses or exams.

During UCLA’s “math bootcamp”, I was presented a solution that involved Cauchy Condensation:

Consider that \(\sum_{n=1}^{\infty}f(n)\leq\int_{0}^{\infty}f(x)dx<\infty\): it’s a right-hand Riemann sum approximating an integral of a nonincreasing function. Cauchy condensation gives \(2^{n}f\left( 2^n \right)\to 0\). Then, for any \(x>1\), pick \(n\) so that \(2^n< x<2^{n+1}\). We then get \(f(x)< f \left( 2^{n} \right)\), so \[xf(x)<2^{n+1}f\left( 2^n \right)=2\cdot 2^{n}f\left( 2^n \right).\] As \(x\to\infty\), \(n\to\infty\) as well, so \(xf(x)\to 0\). Very clean.