Proof That in a Hausdorff Space Every Convergent Sequence Has a Unique Limit
Theorem
If
is a Hausdorff space, then every convergent sequence in
has a unique limit.
Proof
Let
be a convergent sequence inwith limits
Sinceis a Hausdorff space open sets
and
exist such that
Sinceconverges to
Sinceconverges to
Let
then 
and
so
- a contradiction since
Hence every convergent sequence inhas a unique limit.
