Let X be a metric space and let E be a subset of X. Prove that E' is closed and that E and E-closure have the same set of limit points.
The Attempt at a Solution
I have proven that E' is closed. Now, from the definition of a closed subset of a metric space, I know that if E' is closed then every point of E' is a limit point. I also know that E-closure = E ∪ E'. Then, is it right to conclude that (E-closure)' = E' ∪ (E')' = E' ?