Proof That a Convergent Sequence in a Metric Space Has a Unique Limit

Theorem

Ifis a convergent sequence in a metric spacewiththenis unique.

Proof

Suppose conversely thatis a convergent sequence with two distinct limits, so that

and

Since

From the triangle inequality

Sinceconverges to x there existssuch that for alland sinceconverges tothere existssuch that for all

Takethen for alland

Hence

This is an obvious contradiction so every convergence sequence has a unique limit.