---
tag: 9V9AZ10WD1
kind: proof
name: Proof that complete subspaces are closed
msc: ["54E50"]
proves: 0W41XG4W60
uses: [063W5JMY85, 08EWT7J0SC, 0T2WYDTRP2, 0G5AV9Q4RR, 0DRKT1QJR7, 003QP7A8HG, 0D2W7FS52T, 0SHC6Z304Z, 0MCK8K2D7Z, 06JYPJ4WHH, 01M6CW9VM6]
sources:
  - url: https://proofwiki.org/wiki/Subspace_of_Complete_Metric_Space_is_Closed_iff_Complete
    locator: Sufficient Condition
---

## Proof

We show that $A$ equals its closure.

1. Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $A \subseteq X$. Assume $A$ with the [[subspace metric|0T2WYDTRP2]] $d_A$ is [[complete|08EWT7J0SC]].
2. Let $x$ be a point of the [[closure|0DRKT1QJR7]] $\overline{A}$.
3. Since the [[closure is the set of limits of sequences|003QP7A8HG]], there is a sequence $(a_n)$ of points of $A$ that [[converges|06JYPJ4WHH]] to $x$ in $X$.
4. Let $\varepsilon > 0$. There is $N$ with $d(a_n, x) < \varepsilon / 2$ for $n \ge N$.
5. For $m, n \ge N$, the [[triangle inequality|01M6CW9VM6]] gives $d_A(a_m, a_n) = d(a_m, a_n) \le d(a_m, x) + d(x, a_n) < \varepsilon$.
6. So $(a_n)$ is a [[Cauchy sequence|0MCK8K2D7Z]] in $(A, d_A)$. Since $A$ is complete, $(a_n)$ converges in $A$ to some $a \in A$.
7. Since $d(a_n, a) = d_A(a_n, a)$, the sequence $(a_n)$ also converges to $a$ in $X$.
8. 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 $\varepsilon > 0$.
9. So $d(x, a) = 0$. A metric is positive on distinct points, so $x = a$, and thus $x \in A$.
10. By steps 2 to 9, $\overline{A} \subseteq A$. Also $A \subseteq \overline{A}$, because the [[closure is the smallest closed superset|0SHC6Z304Z]].
11. So $\overline{A} = A$. Since [[a set is closed iff it equals its closure|0D2W7FS52T]], $A$ is [[closed|0G5AV9Q4RR]] 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 $A$.