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Math Help - limit points

  1. #1
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    limit points

    Prove that a set  S \subset \bold{R} is closed if and only if it contains all of its limit points.

    Suppose that  S is closed, and  x_0 is a limit point of  S . Then  S^{C} is open so that there is an  \varepsilon > 0 such that  (x_{0} - \varepsilon, x_{0} + \varepsilon) \cap S = \emptyset which is a contradiction.

    If  S  contains all its limit points, then there exists an  \varepsilon > 0 such that  (x_{0}- \varepsilon, x_{0} + \varepsilon) \cap S \neq \emptyset . So  S^{C} is open  \implies S is closed.

    Is this correct?
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  2. #2
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    You have the correct ideas and concepts but an awful form.
    The first part should go something like this. Say that x_0 is limit point of of the closed set S. Suppose x_0  \notin S then x_0  \in S^c which is an open set. So \left( {\exists \delta  > 0} \right)\left[ {\left( {x_0  - \delta ,x_0  + \delta } \right) \subseteq S^c } \right]. But that is a contradiction to x_0 being a limit point.

    No you try to rewrite the second part.
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  3. #3
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     S contains all of its limit points. Suppose that  x_0 \not \in S . Then   x_{0} \in S^{c} , then  (\exists \delta >0)[(x_0- \delta, x_0+\delta) \subseteq S^{c}] . Thus  S^{c} is open which implies that  S is closed.
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  4. #4
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    That looks good.
    You might add that x_0  \in S^c means that x_0 is not a limit point of S.
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