Question

Analysis help @jerico and others.

Answer 1

I assume you are working in the real numbers \(\mathbb{R}\), right? Have you heard of the Heine-Borel theorem ( https://en.wikipedia.org/wiki/Heine-Borel_theorem)? It states that every compact subset of \(\mathbb{R}\) is closed and bounded. Closed and bounded sets contain their glb which should then help you to prove the rest of it, see http://math.stackexchange.com/questions/75608/does-a-closed-and-bounded-set-in-mathbbr-necessarily-contain-its-supremum-a

Maybe this proof is more complicated than necessary, though. There may be a much simpler proof that I just cannot see right now. ;-)