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 .

Proof sketch (optional): If $F$ is closed, then $X\setminus F$ is open , so a limit of points in $F$ cannot lie outside $F$. Conversely, if $F$ contains limits of sequences from $F$ and $x\in\overline{F}$, pick $x_n\in F$ with $x_n\to x$ (closure characterization ); then $x\in F$.