Completeness and closedness - Imported Knowlpedia Content

Proposition. Let $(X,d)$ be a metric space.

If $E\subset X$ is complete, then $E$ is closed in $X$ .
If $X$ is complete and $E\subset X$ is closed, then $E$ is complete (with the restricted metric).

Remarks

Context. This gives a practical way to recognize complete sets: inside a complete ambient space, "closed" and "complete" coincide.

Proof sketch.

Take a sequence $(x_n)\subset E$ converging in $X$ to $x$ . Any convergent sequence is Cauchy, so $(x_n)$ is Cauchy in $E$ . By completeness of $E$ , it converges in $E$ to some $y\in E$ . By uniqueness of limits, $x=y\in E$ , hence $E$ is closed.
Let $(x_n)$ be Cauchy in $E$ . Then it is Cauchy in $X$ and hence converges in $X$ (since $X$ is complete) to some $x\in X$ . Because $E$ is closed and $(x_n)\subset E$ converges to $x$ , we have $x\in E$ . Thus $(x_n)$ converges in $E$ .