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Thread: Bounds

  1. #1
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    Bounds

    1. Let $\displaystyle A $ be a nonempty set of real numbers which is bounded below. Let $\displaystyle -A $ be the set of numbers $\displaystyle -x $, where $\displaystyle x \in A $. Prove that $\displaystyle \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; Feb 16th 2008 at 02:08 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, $\displaystyle k = \inf (A)$.
    Thus, $\displaystyle x \in A\quad \Rightarrow \quad k \leqslant x\quad \Rightarrow \quad - x \leqslant - k$.
    That means that $\displaystyle -k$ is an upper bound for $\displaystyle -A$ so let $\displaystyle j = \sup ( - A)$.
    We know that $\displaystyle j \leqslant - k$ so suppose that $\displaystyle j < - k$.
    Then $\displaystyle k < - j\quad \Rightarrow \quad \left( {\exists t \in A} \right)\left[ {k \leqslant t < - j} \right]\,or\,j < - t \leqslant - k$

    Do you see a contradiction?
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