Proof That Every Cauchy Sequence in a Metric Space is Bounded and Totally Bounded
Every Cauchy sequencein a metric spaceis bounded and totally bounded.
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