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Math Help - Bounds

  1. #1
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    Bounds

    1. Let  A be a nonempty set of real numbers which is bounded below. Let  -A be the set of numbers  -x , where  x \in A . Prove that  \inf(A) = -\sup(-A) .

    Intuitively this makes sense if you draw it on a number line. But I am not sure how to formally prove it.
    Last edited by heathrowjohnny; February 16th 2008 at 03:08 PM.
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  2. #2
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    does it just tautologically follow from the definitions?
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  3. #3
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    We know that A has a greatest lower bound, k = \inf (A).
    Thus, x \in A\quad  \Rightarrow \quad k \leqslant x\quad  \Rightarrow \quad  - x \leqslant  - k.
    That means that -k is an upper bound for -A so let j = \sup ( - A).
    We know that j \leqslant  - k so suppose that j <  - k.
    Then k <  - j\quad  \Rightarrow \quad \left( {\exists t \in A} \right)\left[ {k \leqslant t <  - j} \right]\,or\,j <  - t \leqslant  - k

    Do you see a contradiction?
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