RMM 2009 Problem 4

For a finite set $X$ of positive integers, let $\Sigma(X) = \sum_{x \in X} \arctan \frac{1}{x}.$ Given a finite set $S$ of positive integers for which $\Sigma(S) < \frac{\pi}{2},$ show that there exists at least one finite set $T$ of positive integers for which $S \subset T$ and $\Sigma(S) = \frac{\pi}{2}.$ Kevin Buzzard, United Kingdom