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Limit point proof

  1. Oct 12, 2008 #1
    1. The problem statement, all variables and given/known data
    I am trying to prove that every uncountable subset of R has a limit point in R.

    (where R is the reals)

    2. Relevant equations
    I know that the reals are dense and you can find a real in between any two reals. I feel like if you keep finding a real in between two reals, you will find a limit point. I am not even sure where to start with this idea.

    3. The attempt at a solution
  2. jcsd
  3. Oct 12, 2008 #2
    Start with the definition of limit point. What is it?
  4. Oct 12, 2008 #3
    Limit point of a set A: A limit point is a point x in which any neighborhood centered at x, no matter the size, intersects the set A at a point other than x.

    The Bolzano-Weierstrass thrm states that , every bounded infinite subset of R has a limit point.

    My problem does not have the bounded part, and is uncountable as opposed to the puny infinite.

    I really don't understand why this works. Okay, if it's bounded and infinite then I feel convinced that it has a limit point (after seeing the thrm, of course). But non bounded, and uncountable? Is this problem stronger than the BW theorem?
  5. Oct 12, 2008 #4
    Try using contradiction. Suppose your uncountable subset A in R does not have a limit point. That means, for any x in A, there is some ball around x which does not contain any element of A except itself. This should lead to a contradiction of A being uncountable.
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