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

*Let be the set of all limit points of the set . Prove that 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

. if then is not a limit point of so there exist a neighborhood such that for some . Therefore, we have that is a limit point of .

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