Proof that complete subspaces are closed

Proof

We show that $A$ equals its closure.

Let $(X, d)$ be a metric space and let $A \subseteq X$ . Assume $A$ with the subspace metric $d_A$ is complete.
Let $x$ be a point of the closure $\overline{A}$ .
Since the closure is the set of limits of sequences, there is a sequence $(a_n)$ of points of $A$ that converges to $x$ in $X$ .
Let $\varepsilon > 0$ . There is $N$ with $d(a_n, x) < \varepsilon / 2$ for $n \ge N$ .
For $m, n \ge N$ , the triangle inequality gives $d_A(a_m, a_n) = d(a_m, a_n) \le d(a_m, x) + d(x, a_n) < \varepsilon$ .
So $(a_n)$ is a Cauchy sequence in $(A, d_A)$ . Since $A$ is complete, $(a_n)$ converges in $A$ to some $a \in A$ .
Since $d(a_n, a) = d_A(a_n, a)$ , the sequence $(a_n)$ also converges to $a$ in $X$ .
For each $n$ , $d(x, a) \le d(x, a_n) + d(a_n, a)$ . The right side tends to $0$ , so $d(x, a) < \varepsilon$ for every real $\varepsilon > 0$ .
So $d(x, a) = 0$ . A metric is positive on distinct points, so $x = a$ , and thus $x \in A$ .
By steps 2 to 9, $\overline{A} \subseteq A$ . Also $A \subseteq \overline{A}$ , because the closure is the smallest closed superset.
So $\overline{A} = A$ . Since a set is closed iff it equals its closure, $A$ is closed in $X$ .

Idea

A point of the closure is a limit of points of $A$ . Such a sequence is Cauchy, so completeness of $A$ gives it a limit inside $A$ . Limits are unique, so the original point is that limit and lies in $A$ .

Tag	`9V9AZ10WD1`
URI	`m2://9V9AZ10WD1`
Kind	proof
Writing	W0 stub
MSC	54E50
Proves	Complete subspaces are closed
Uses	Metric space, Complete metric space, Subspace metric, Closed set, Closure, Closure is the set of limits of sequences, A set is closed iff it equals its closure, Closure is the smallest closed superset, Epsilon characterisation of Cauchy sequences, Convergence of a sequence in a metric space, Triangle inequality for a metric
Changed	4 November 2026

Tag

9V9AZ10WD1

URI

m2://9V9AZ10WD1

Kind

proof

Writing

W0 stub

MSC

54E50

Proves

Complete subspaces are closed

Uses

Metric space, Complete metric space, Subspace metric, Closed set, Closure, Closure is the set of limits of sequences, A set is closed iff it equals its closure, Closure is the smallest closed superset, Epsilon characterisation of Cauchy sequences, Convergence of a sequence in a metric space, Triangle inequality for a metric

Changed

4 November 2026

Proof that complete subspaces are closed

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