Proof That Sequentially Compact Subsets of a Metric Space are Totally Bounded

Theorem

Sequentially compact subsets of a metric space are totally bounded.

Proof

Letbe a subset of a metric spaceSupposeis not totally bounded, thenexists such that no- net exists. Let

exists such thatotherwisewould be- net.

exists such thatotherwisewould be- net.

Continue this procedure to obtain a set of pointssuch that for

This sequence does not have a convergent subsequence, henceis not sequentially compact.

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