Compactness in Metrizable Spaces: Proving X is Compact with Bounded Metrics

[tylerc1991]

Homework Statement

Let (X,T) be a metrizable space such that every metric that generates T is bounded. Prove that X is compact.

The Attempt at a Solution

I was thinking about this problem a bit before I headed off to work and wanted to get you guys' thoughts and/or ideas. At first I was trying to use the original definition of compactness, i.e. letting O be some open cover of X and assuming that there is no finite subcover of X and arriving at a contradiction. I didn't get anywhere with this so then I thought about trying to show sequential compactness but I don't see how a sequence necessarily converges in X. Any ideas? Thank you for your help!

 

[micromass]

Not sure if this approach will work, but you should try it out. Assume it is not compact, so you can find a sequence $$(x_n)_n$$ without convergent subsequence. Then $$X\setminus \{x_n~\vert~n>0\}$$ is an open set which is metrizable by a bounded metric (say d(x,y)<1). Now, we adjoin x_1 to this set and we set the distance d'(x_1,y)=2. Then we adjoin x_2 and we set the distance d'(x_2,y)=3, and so on.

I have a feeling this should work, but there are some details which you still need to check...