Hausdorff implies US

Statement
Any Hausdorff space is a US-space: every convergent sequence has a unique limit.

Related facts

 * US not implies Hausdorff
 * T1 not implies US
 * US implies T1

Proof
Given: A Hausdorff space $$X$$. A convergent sequence $$x_1, x_2, \dots $$ with limit $$x$$ and limit $$y$$.

To prove: $$x = y$$.

Proof: Suppose $$x \ne y$$. Then, by Hausdorffness of $$X$$, there exist disjoint open subsets $$U$$ and $$V$$ such that $$x \in U, y \in V$$.

The definition of limit tells us that for all but finitely many $$i$$, $$x_i \in U$$, and for all but finitely many $$i$$, $$x_i \in V$$. Hence, there exists a value of $$n$$ for which $$x_n \in U \cap V$$, which is a contradiction since $$U$$ and $$V$$ are disjoint. This completes the contradiction, and hence the proof.