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## Homework Statement

Let ##E## be a metric subspace to ##M##. Show that ##E## is closed in ##M## if ##E## is complete. Show the converse if ##M## is complete.

## Homework Equations

A set ##E## is closed if every limit point is part of ##E##.

We denote the set of all limit points ##E'##.

A point in ##p\in M## is a limit point to ##E\subseteq M## if ##\forall \epsilon > 0## ##\exists q \in E \cap B(p,\epsilon)##

## The Attempt at a Solution

We want to show that ##E' \subseteq E##. Take ##p \in E'## then clearly we can choose

##p_n \in B(p,1/n)## so that ##p_n \in E##.

But then for all ##\epsilon > 0## ##d(p_n,p_m)\le d(p_n,p)+ d(p_m,p)<\epsilon## for ##n,m \ge N_\epsilon = 2/\epsilon## i.e. ##(p_n)## is a cauchy sequence.

But then it must converge to some ##\lambda \in E##. However ##\lambda = p## since ##d(p,\lambda) = d(p_n,p)+d(p_n,\lambda)< \epsilon##

Is this a correct proof? Should I use a similar approach to the second part?