Closure is the set of limits of sequences

Statement

Let $(X, d)$ be a metric space, let $S \subseteq X$ and let $x \in X$. Then $x$ is in the closure $\overline{S}$ if and only if there is a sequence $(s_n)$ of points of $S$ that converges to $x$.

Remarks

Proof of the direct part: by closure in a metric space, for each $n \ge 1$ there is $s_n \in S$ with $d(x, s_n) < 1/n$. Then $s_n \to x$. The converse follows because $\overline{S}$ is closed and closed sets contain their limits. A consequence: $S$ is closed if and only if the limit of each convergent sequence of points of $S$ is in $S$. In a general topological space, sequences are not enough, and one uses nets or filters instead.

Explanations

L1 human

In a metric space, the closure of a set consists of the limits of sequences that lie in the set. For example, each real number is a limit of rational numbers, so the closure of $\mathbb{Q}$ is $\mathbb{R}$.