Uniqueness of limits in Hausdorff spaces

Uniqueness of limits in Hausdorff spaces: Let $X$ be a Hausdorff space and let $(x_n)$ be a sequence in $X$. If $(x_n)$ converges to both $x\in X$ and $y\in X$, then $x=y$.

This property is a key reason Hausdorff spaces behave like metric spaces with respect to convergence, and it is closely aligned with results like compact subsets are closed .