- #1
tylerc1991
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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!