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Math Help - Bounded Set With Exactly 3 Limit Points

  1. #1
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    Bounded Set With Exactly 3 Limit Points

    Hello, the problem I have is:

    Construct a bounded set of real numbers with exactly three limit points.

    I have what I believe to be a solution:

    A = {1/x: x E N}U{1 + 1/x: x E N}U{2 + 1/x: x E N} => A is bounded by 3 with limit points 0, 1, 2.

    Unfortunately, I don't really understand the concept of a limit point. My text defines a limit point as "a point p is a limit point of the set E if every neighborhood of p contains a point q (not equal) p such that q is in E"

    Can anyone explain this concept to me or tell me if this solution is correct? Thank you in advance.
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  2. #2
    A Plied Mathematician
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    Your solution works fine (there's loads of solutions). The idea of a limit point p is that there are lots and lots (infinitely many, as you can prove) of distinct points in E that get arbitrarily close to p. In fact, it doesn't matter how much you "zoom in" on the point p, there will still be infinitely many points in your "viewing window". Does that make sense?
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  3. #3
    MHF Contributor

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    Another way of thinking about a "limit point" of a set is that there is a sequence (with an infinite number of distinct values) in the set having the point as its limit.

    So take three different sequences in R, converging to three different numbers. Their union will be a sequence having those three points as subsequential limits and so will be a set having 3 different limit points.
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