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Proof That Every Cauchy Sequence in a Metric Space is Bounded and Totally Bounded

Theorem

Every Cauchy sequencein a metric spaceis bounded and totally bounded.

Proof

Letbe a Cauchy sequence in a metric space

A subsetof a metric spaceis totally bounded ifhas an- net for every (an- net foris a finite set of pointssuch that for everythere issuch that).

LetSinceis Cauchy, there existssuch that

Hence the diameter of the setis at mostThe- net for the sequenceis then