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Math Help - accumulation points

  1. #1
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    accumulation points

    how can you prove this theorem from the book?

    A finite set has no accumulation points.

    I know that it's true, but fail to show it mathematically
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  2. #2
    A Plied Mathematician
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    Are you working in the real numbers? What ideas have you had so far?
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  3. #3
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by EmmWalfer View Post
    how can you prove this theorem from the book?

    A finite set has no accumulation points.

    I know that it's true, but fail to show it mathematically

    Well, if you construct an epsilon neighborhood around a proposed limit point x in your finite set, then the intersection of your set with the epsilon neighborhood of x must contain points in your finite set other than x for x to be a limit point (accumulation point). If you can show this doesn't hold, then you are done.
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  4. #4
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    The definition of "accumulation point" is: p is an accumulation point of set A if and only if every neighborhood of p contains at least one point of A (other than p itself). Suppose A is finite. Then the set of all distances from p to points in A (other that p itself) is finite and so contains a smallest value. Take the radius of your neighborhood to be smaller than that value.

    (I notice now, that is pretty much what Danneedshelp said!)
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  5. #5
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    The definition of "accumulation point" is: p is an accumulation point of set A if and only if every neighborhood of p contains at least one point of A (other than p itself). Suppose A is finite. Then the set of all distances from p to points in A (other that p itself) is finite and so contains a smallest value. Take the radius of your neighborhood to be smaller than that value.

    (I notice now, that is pretty much what Danneedshelp said!)
    I am not a 100% this correct, but I recall a lemma from my intro to real analysis book that stated something along to lines of: a point x is a limit point (accumulation point) of a set A iff for every \epsilon>0, the neighborhood N_{\epsilon}(x) contains infinitly many points of the set A.

    I think you would have to prove this to use it, but it answers your original question rather quickly.
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