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Proof That a Sequentially Compact Space is Countably Compact

Theorem

A sequentially compact set is countably compact.

Proof

Supposeis a sequentially compact space. Letbe any infinite subset of

We can find a sequenceinwith distinct terms, so that for

Sinceis sequentially compactcontains a subsequence which converges to a point inThen x is an accumulation point ofandis countably compact.