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Thread: Sup(S) proof

  1. #1
    Sep 2009

    Sup(S) proof

    S is a nonempty set in the real numbers. Prove:
    M = sup(S) iff M is an upper bound of S and there exists a sequence (call it a_n) such that lim(a_n) = M as n goes to infinity.
    Having trouble with this one, any suggestions?
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  2. #2
    Junior Member
    Apr 2009
    Here's one direction..

    If M is an upper bound and this sequence exists, then for any epsilon > 0 there exists an N such than for all n > N,

    M - a_n < epsilon

    Assume B is a lesser upper bound than M. Then take epsilon = M - B and we get a_n > B for n large enough. But this contradicts B being an upper bound. Hence M is the least upper bound.
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