prove that every compact metric space is complete.
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A metric space is sequentially compact if and only if it is complete and totally bounded. Proof. Suppose that X is a sequentially compact metric space. Then every Cauchy sequence in X has a convergent subsequence, so, by Lemma 6 below, the Cauchy sequence converges, meaning that X is complete.
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Step-by-step explanation:
A metric space is sequentially compact if and only if it is complete and totally bounded. Proof. Suppose that X is a sequentially compact metric space. Then every Cauchy sequence in X has a convergent subsequence, so, by Lemma 6 below, the Cauchy sequence converges, meaning that X is complete.
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