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Advanced Calculus proof

  1. Dec 12, 2006 #1
    Prove that a is a cluster point of E if and only if the set (E intersection (a-r,a+r))\{a} is nonempty for each r > 0.

    I have the forward implication done but the backwards implication is giving me some trouble. Could you explain it to me.
     
  2. jcsd
  3. Dec 12, 2006 #2

    StatusX

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    What is your definition of "cluster point?"
     
  4. Dec 12, 2006 #3
    A point a in the reals is called a cluster of E if (E intersection(a-r,a+r) contains infinitely many points for every r>0.
     
  5. Dec 12, 2006 #4

    Hurkyl

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    Well, if you're having trouble proving something complicated, try proving something simple first.

    "infinitely many points" is a lot... maybe you can prove that

    if the set (E intersection (a-r,a+r))\{a} is nonempty for each r > 0​

    then

    (E intersection(a-r,a+r) contains one point for every r>0.​

    Then what about two points? Three points?
     
  6. Dec 13, 2006 #5

    StatusX

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    If there are only finitely many points in (a-r,a+r)-{a}, then there is some point closest to a, and so...
     
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