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- I am reading a proof of the fact that if ##E\subset\mathbb R^n## 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 ##X## is totally bounded if for each ##\epsilon>0##, there exists a finite number of open balls of radius ##\epsilon>0## that cover ##X##.
Question: How can I verify that the balls ##B(\epsilon j,\epsilon)## cover ##T##? In particular, why the condition ##\epsilon |j_i|\leq 2b##, ##1\leq i\leq n##?
Proof: Since ##E## is bounded, it is contained in some cube ##T=[-b,b]^n## and it suffices to show that ##T## is totally bounded since a subspace of a totally bounded metric space is totally bounded. For this, let ##\epsilon>0##. Then the balls ##B(\epsilon j,\epsilon)## cover ##T##, where ##j=(j_1,\ldots,j_n)## ranges over all integral lattice points of ##\mathbb R^n## which satisfy ##\epsilon |j_i|\leq 2b##, ##1\leq i\leq n## (recall, an integral lattice point is a point whose coordinates are integers). Since there are only finitely many such lattice points, ##T## is totally bounded.
Question: How can I verify that the balls ##B(\epsilon j,\epsilon)## cover ##T##? In particular, why the condition ##\epsilon |j_i|\leq 2b##, ##1\leq i\leq n##?