Proof That a Convergent Sequence in a Metric Space Has a Unique Limit
Ifis a convergent sequence in a metric spacewiththenis unique.
Suppose conversely thatis a convergent sequence with two distinct limits, so that
From the triangle inequality
Sinceconverges to x there existssuch that for alland sinceconverges tothere existssuch that for all
Takethen for alland
This is an obvious contradiction so every convergence sequence has a unique limit.