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.

Proof sketch (optional): If $x\in\overline{E}$, then every ball $B(x,1/n)$ meets $E$; choose $x_n\in E\cap B(x,1/n)$ to get $x_n\to x$. Conversely, if $x_n\to x$ with $x_n\in E$, then every neighborhood of $x$ contains some $x_n$, so it meets $E$ and $x\in\overline{E}$.