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Thread: Prove sup S <= inf T

  1. #1
    Junior Member
    Feb 2010

    Prove sup S <= inf T

    Let S and T be non-empty subsets of R, and suppose that for all s ε S and t ε T, we have s <= t. Prove that supS <= infT.

    Here's what I have for it:

    Since s ε S and s ε T, supT is an upper bound for S.
    Since supS is the least upper bound, sup S <= sup T.

    How does that look? Help is greatly appreciated. Thanks.
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  2. #2
    Senior Member Tinyboss's Avatar
    Jul 2008
    Suppose the contrary, that sup S > inf T. Let $\displaystyle \epsilon=(\sup S-\inf T)/2$, and follow the definitions of sup and inf to derive a contradiction.
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