Lemma: Characterization of Closed Sets by Limits of Sequences

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

Proofs: 1 2

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  1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984