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Math Help - Real Analysis: Limit Point

  1. #1
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    Real Analysis: Limit Point

    Q: Suppose that A and B are non-empty sets of real numbers and that x is a limit point of AUB. Prove that X is a limit point of A or of B.

    my solution:

    Given x is a limit point of AUB

    then for all epsilon > 0, nbd(x) contains a point of AUB different from x


    Then nbd(x) contains a point of A or nbd(x) contains a point of B different from x.


    Therefore x is a limit point of A or of B.


    [nbd = neighborhood]

    Just wondering if my approach is correct or not. I feel it's not as easy as I thought...

    help please thanks!
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  2. #2
    MHF Contributor

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    You are correct, that is not sufficient to prove it.
    Suppose that x is not a limit point of A.
    Then some neighborhood, N(x) contains no point of A-\{x\}.

    But for any neighborhood, M(x) then Q(x)=M(x)<br />
\cap N(x) is also a neighborhood of x which must contain points of (A\cup B)}-\{x\}.

    Does that make x a limit point of B?
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