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Math Help - Relative supremum proof

  1. #1
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    Relative supremum proof

    If sup B > sup A, then prove that there is an element b of B that is an upper bound for A.

    My work: I split this into two cases, that where B has a maximum, and that where it doesn't. If it has a maximum, then we can set b = sup B = max B, and clearly this is an upper bound for A since sup B > sup A. But I'm lost for the second case...

    In general, I'm having trouble with these supremum proofs when it comes to the case where the set doesn't have a maximum. Any help?
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  2. #2
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    sup A can't be an upper bound for B anymore, because it is smaller than sup B (which is already the SMALLEST upper bound of B).
    So therefore NOT all b are <= sup A.
    =>
    There exists a b , with b>sup A.
    sup A is an upper bound of A, so b must also be an upper bound of A.
    Last edited by Iondor; August 25th 2010 at 05:30 AM.
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  3. #3
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    In your second case, you know that there are b arbitrarily close to sup(B), so then there must be a b such that sup(B) > b > sup(A), for if sup(B) > sup(A) > b for all b, then we cannot find b's that are arbitrarily close to sup(B).
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