Sequential Characterization of Closed Subsets of 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 Metric Open Sets Form a Topology. Let $A\subseteq X$, and let $\mathbb{N}$ denote the natural numbers.

Then $A$ is closed in the topological space $(X,\mathcal{T}_d)$ if and only if the following condition holds: for every sequence $(x_m)_{m\in\mathbb{N}}$ in $X$ such that $x_m\in A$ for every $m\in\mathbb{N}$, and every point $x\in X$ such that $(x_m)_{m\in\mathbb{N}}$ converges to $x$ in $(X,d)$, one has $x\in A$.