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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.