Theorem: Let $\displaystyle X$ be Hausdorff, then if $\displaystyle \{x_n\}_{n\in\mathbb{N}}$ is a convergent sequence in $\displaystyle X$, then it's limit $\displaystyle x$ is unique.
Proof: Suppose that $\displaystyle x\ne y$ and let $\displaystyle U_x,U_y$ be the respective disjoint neighborhoods of $\displaystyle x,y$ guaranteed by Hausdorffness. Now, since $\displaystyle x_n\to x$ all but finitely elements of $\displaystyle K=\{x_n:n\in\mathbb{N}\}$ lie in $\displaystyle U_x$ and so not all but finitely man values of $\displaystyle K$ may lie in $\displaystyle U_y$. It follows that $\displaystyle x_n\not\to y$. The conclusion follows.
Soo..................