Let $(X, \mathcal{T})$ be [[Hausdorff topological spaces|Hausdorff space]] and $(x_{n})_{n=1}^{\infty}$ a [[Convergence|convergent sequence]] in $X$. Then its limit is unique **Proof**: Suppose $x_{n}\to x$ and $x_{n}\to y$. If $x \neq y$, then there exist $U, V\in \mathcal{T}$ containing $x,y$ respectively such that $U \cap V = \emptyset$. But this contradicts the fact that there exists $N, M$ such that for all $n \geq \text{max}\{N, M\}$, $x_{n}\in U$ and $x_{n}\in V$.