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Math Help - Supremum Problem

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    Supremum Problem

    Show that sup{R element Q: r < a} = a for each a element R.
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  2. #2
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    Quote Originally Posted by tbyou87 View Post
    Show that sup{R element Q: r < a} = a for each a element R.
    Thanks
    These are "Dedekind Cuts".

    We need to show:
    1)"a" is an upper bound.
    2)"a" is least upper bound.

    #1)It is an upper bound, for that is the definition of this non-empty.

    #2)Say that "a" is not a least upper bound say b<a is the least upper bound. Then by the denseness of Q we have that there exists an rational "r" such that b<r<a hence, r in the set. But then "b" cannot be an unpper bound. A contradiction. Thus "a" is the least upper bound.
    Q.E.D.
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