Question: A Cauchy sequence is convergent it has a convergent subsequence. Actually compact spaces have a much stronger property than boundedness. A metric space X is

A Cauchy sequence is convergent ‡” it has a convergent subsequence.
Actually compact spaces have a much stronger property than boundedness. A metric space X is totally bounded if, for every r > 0, it is contained in a finite number of open ball Br(xi) of radius r, that is,

The open balls are said to cover X.

X = U B,(x) i=1

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