Closure of a Set in a Metric Space is all its Sequential Limits
# Statement
Suppose $(X, d)$ is a Metric Space. Suppose $S \subset X$ Then $$\text{cl} S = {x : \exists (x_n) \subset S \text{ s.t. } x_{n} \to x}$$
# Proof
Metric Spaces are First Countable and Hausdorff and Closure of a Set in a First Countable Space is all its Sequential Limits. $\blacksquare$