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psie
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- TL;DR Summary
- I am reading a proof of the fact that if
is bounded, then it is totally bounded. This proof is from Gamelin's and Greene's book on topology. I have a question about the proof.
Recall, a set is totally bounded if for each , there exists a finite number of open balls of radius that cover .
Question: How can I verify that the balls cover ? In particular, why the condition , ?
Proof: Sinceis bounded, it is contained in some cube and it suffices to show that is totally bounded since a subspace of a totally bounded metric space is totally bounded. For this, let . Then the balls cover , where ranges over all integral lattice points of which satisfy , (recall, an integral lattice point is a point whose coordinates are integers). Since there are only finitely many such lattice points, is totally bounded.
Question: How can I verify that the balls