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Math Help - Proof of how multiplying a positive and negative constant changes the sup and inf

  1. #1
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    Proof of how multiplying a positive and negative constant changes the sup and inf

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    Last edited by tempins; August 10th 2010 at 12:46 AM. Reason: solved
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  2. #2
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    For your second question, it's more along those lines. Let A = \inf(S) and B= \sup(S) where A < B. Let a be a positive number. Then -a is a negative number.

    -aA > -aB by the rules of ">".

    Hence now you have that \inf(S) = -aB and  \sup(S) = -aA

    Try an example: the interval (1,2) for example. A = 1, B = 2. When you take [LaTeX ERROR: Convert failed] , you get  (-2,-1) and it should be clear.
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    Please do NOT erase a post just because you do not need it any more. New people looking at the thread can still learn something from it.
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  4. #4
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    The question was along the lines of:

    If S is a set and S \subset \mathbb{R} and if a \in \mathbb{R} is a constant, then let aS denote the set aS = \{ ax \mid x \in S\}.

    Then if A = \inf(S) and B = \sup(S), prove:

    If a > 0 then \inf(aS) = a\inf(S) and \sup(aS) = a\sup(S).
    If a < 0 then \inf(aS) = a\sup(S) and \sup(aS) = a\inf(S).
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