Sequential Characterization of the Closure in a Metric Space

Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of all subsets of $X$ that are open in $(X,d)$, which is a topology on $X$ by The Open Subsets of a Metric Space Form a Topology. Let $A\subseteq X$, let $x\in X$, and let $\mathbb{N}$ denote the natural numbers.

Then $x$ belongs to the closure $\operatorname{cl}_X(A)$ of $A$ in $(X,\mathcal{T}_d)$ if and only if there exists a sequence $(a_m)_{m\in\mathbb{N}}$ in $X$ such that $a_m\in A$ for every $m\in\mathbb{N}$ and such that $(a_m)_{m\in\mathbb{N}}$ converges to $x$ in $(X,d)$.