## Proof That a Convergent Sequence in a Metric Space is Cauchy

Theorem

Every convergent sequencein a metric spaceis Cauchy.

Proof

Letbe a convergent sequence in a metric spaceso thator as

Then for everythere existssuch that

Hence

Henceis Cauchy.

Theorem

Every convergent sequencein a metric spaceis Cauchy.

Proof

Letbe a convergent sequence in a metric spaceso thator as

Then for everythere existssuch that

Hence

Henceis Cauchy.