Proof That a Sequentially Compact Subset of a Metric Space is Compact

Theorem

A sequentially compact subsetof a metric spaceis compact.

Proof

Letbe a sequentially compact subset of a metric spaceand letbe an open cover ofThe sethas a Lebesgue numbersince every open cover of a sequentially compact subset of a metric space has a Lebesgue number. Sinceis totally bounded, there is a decomposition ofinto a finite number of subsetswith whereis the diameter of the subset

is a Lebesgue number ofso there are open setssuch that

Henceandis a finite subcover. Henceis compact.

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