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Math Help - Limit Point Proof.

  1. #1
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    Limit Point Proof.

    Suppose that A and B are non-empty sets of real numbers, and that x is a limit point of A union B. Prove that x is a limit point of A or of B.


    => I think that I need to do proof by cases for this one, and use the epsilon neighborhood in some way, but I am not sure.

    Also, could this be more easily proved using proof by contradiction,
    assuming that x is not a limit point of a and x is not a limit point of B.

    Thanks for you help.
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  2. #2
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    Quote Originally Posted by eg37se View Post
    Suppose that A and B are non-empty sets of real numbers, and that x is a limit point of A union B. Prove that x is a limit point of A or of B.


    => I think that I need to do proof by cases for this one, and use the epsilon neighborhood in some way, but I am not sure.

    Also, could this be more easily proved using proof by contradiction,
    assuming that x is not a limit point of a and x is not a limit point of B.

    Thanks for you help.

    So there exists a sequence {x_n} in A \/ B s.t. x_n --> x . For any n, x_n belongs either to A or to B. It can't be that both A and B contain a finite number of elements of the sequence ==> either A or B (or both, of course, as "or" in mathematics is always inclusive unless otherwise stated) contain an infinite number of elements of {x_n} ==> since {x_n} converges so does any infinite subsequence and to the same limit ==> we're done.

    Tonio
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  3. #3
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    I considered this way too, I just wasn't sure. Thank you for the help.
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