Sequential characterization of closed sets

Sequential characterization of closed sets: Let $(X,d)$ be a metric space and $F\subseteq X$. Then $F$ is closed if and only if whenever $(x_n)$ is a sequence in $F$ with $x_n\to x$ in $X$, one has $x\in F$.

This gives a practical criterion for closedness using sequences, avoiding direct work with complements or open balls .