Suppose that (X, d) is complete and that A is a closed subset. Show A must be complete.
As a beginner with metric Spaces can someone help out with this? I know both the definitions of complete and closed but in applying them I struggle slightly so any help would be so appreciated!
Suppose $\{x_n\}_{n=1}^{\infty}$ be a Cauchy sequence in $A$. As $X$ is complete $\{x_n\}_{n=1}^{\infty}$ must converge to a point $x\in X$. But $A$ is closed it must contain all its limit points. And $x$ is a limit point of $A$, hence $x\in A$. Therefore any Cauchy sequence in $A$ converges to a point in $A$. Hence $A$ is complete.
