## Proof That a Subsequence of a Cauchy Sequence is Cauchy

Theorem

Letbe a Cauchy sequence in a metric spaceand letbe a subsequence.

Thenas

Proof

LetSinceis Cauchy, there existssuch that

Butand

Henceas

Theorem

Letbe a Cauchy sequence in a metric spaceand letbe a subsequence.

Thenas

Proof

LetSinceis Cauchy, there existssuch that

Butand

Henceas