Let $X$ be a metric space. A subset \(A\subset X\) is closed if and only if with any convergent sequence of points $(x_k)_{k\in\mathbb N}$ contained in $A$ also its limit is contained in $A$, formally $$A\text{ is closed }\Longleftrightarrow\text{ for all } (x_k)_{k\in\mathbb N}:~\text{ if }x_k\in A \text{ for }k\in\mathbb N~\text{ then }\lim_{k\to\infty} x_k=x\in A.$$
This is the generalization of the theorem, how convergence of real sequences preserves upper and lower bounds for sequence members.
Proofs: 1
