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Math Help - Closed set

  1. #1
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    Closed set

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

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

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

    Any help or pointers on what I should do next would be greatly appreciated.
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  2. #2
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    There is a similar question here.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by putnam120 View Post
    I am having trouble solving this problem from Principles of Mathematical Analysis by Rudin.

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

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

    Any help or pointers on what I should do next would be greatly appreciated.
    You have gone slightly to far. What you have shown is that for all x \in A there exists a neighbourhood of x which contains no points of E', so is a subset of A, which proves that A is open.

    RonL
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