Every continuous function bounded implies compact

February 18, 2009

It occurs to me that it might be nice to post solutions to miscellaneous mathematical exercises at least once in a while.

I saw this one on a chalkboard earlier today; evidently the room was serving as the venue for an analysis class. It’s exactly the sort of elementary exercise that usually takes me a day to solve, if I’m lucky. But this time, I’m happy to report, I managed to figure it out in just a few minutes (while ostensibly listening to a lecture on something else).

Problem: Let $U \subset \mathbf{R}^n$ be such that every real-valued continuous function on ${}U$ is bounded. Prove that $U$ is compact.