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Math Help - Closed set

  1. #1
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    Closed set

    Hey, I was just wondering if anyone could tell me if this was about right.

    Let E' be the set of limit points of E. Then E' is closed.

    Proof:
    Take an arbitrary limit point of E', say y.

    Then, for any real  \epsilon > 0, there exists a point p in E' such that  p \in N_{\epsilon}(y) .
    But, for any p \in E', there exists a real  \delta , with  0 < \delta < \epsilon such that, for some point q in E,  q \in N_{\delta}(p)

    But then,  q \in N_{\epsilon}(y) too, so y must be a limit point of E', and E' is closed by definition.
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  2. #2
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    Quote Originally Posted by Math Major View Post
    Hey, I was just wondering if anyone could tell me if this was about right.

    Let E' be the set of limit points of E. Then E' is closed.

    Proof:
    Take an arbitrary limit point of E', say y.

    Then, for any real  \epsilon > 0, there exists a point p in E' such that  p \in N_{\epsilon}(y) .
    But, for any p \in E', there exists a real  \delta , with  0 < \delta < \epsilon such that, for some point q in E,  q \in N_{\delta}(p)

    But then,  q \in N_{\epsilon}(y) too, so y must be a limit point of E', and E' is closed by definition.
    In a metric space this is the general idea.
    But you need to take care: \delta=\min\left\{d(p,y),\epsilon-d(p,y)\right\}.
    The insures subsetness.
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