 Solution

A sequence $x_n$ converges to $x$ if every open neighbourhood of $x$ contains all sufficiently late terms. A Hausdorff space separates any two distinct points by disjoint open neighbourhoods.

If a sequence had distinct limits $x,y$ in a Hausdorff space, choose disjoint open neighbourhoods $U$ of $x$ and $V$ of $y$. Convergence to each limit implies that all sufficiently late terms lie in both $U$ and $V$, contradicting $U\cap V=\varnothing$. Hence a convergent sequence in a Hausdorff space has a unique limit.