Proof That in a Hausdorff Space Every Convergent Sequence Has a Unique Limit

Theorem

Ifis a Hausdorff space, then every convergent sequence inhas a unique limit.

Proof

Letbe a convergent sequence inwith limits

Sinceis a Hausdorff space open setsandexist such that

Sinceconverges to

Sinceconverges to

Letthen andso- a contradiction since

Hence every convergent sequence inhas a unique limit.