I am having trouble solving this problem from Principles of Mathematical Analysis by Rudin.

*Let $\displaystyle E'$ be the set of all limit points of the set $\displaystyle E$. Prove that $\displaystyle E'$ is closed.

What I am trying to do is prove that the complement of the set is open thus making the set itself closed. So far I have

$\displaystyle A=(E')^c$. if $\displaystyle x\in A$ then $\displaystyle x$ is not a limit point of $\displaystyle E$ so there exist a neighborhood $\displaystyle N_r(x)$ such that $\displaystyle N_r(x)\subset E^c$ for some $\displaystyle r>0$. Therefore, $\displaystyle \forall x\in A$ we have that $\displaystyle x$ is a limit point of $\displaystyle E^c$.

Any help or pointers on what I should do next would be greatly appreciated.