Let E' be the set of all limit points of a set E. Prove that E' is closed.

Let x be a limit point of E'. So every nbd of x contains infinitely many points of E'. But in every nbd of every point of E' are infinitely many points of E, since every point of E' is a limit point of E.

Thereby every nbd of x contains infinitely many points of E. So x is a limit point of E (ie) x is in E'.

Hence E' is closed. Am i right?