Sequential characterization of closure

Sequential characterization of closure: Let $(X,d)$ be a metric space and $E\subseteq X$. A point $x\in X$ belongs to the closure $\overline{E}$ if and only if there exists a sequence $(x_n)$ in $E$ such that $x_n\to x.$

This result ties topological notions (closure) to analytic ones (sequences) and is one reason sequences are so effective in metric spaces.