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Math Help - Simple supremum question

  1. #1
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    Simple supremum question

    If A and B are bounded and non-empty subsets of \mathbb{R}, and c \in \mathbb{R}, let

    cA = { ca: a \in A}.

    Prove the following:
    if c > 0, supremum cA = c supremum A.

    [SOLVED]
    Last edited by cgiulz; September 29th 2009 at 02:15 PM.
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  2. #2
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by cgiulz View Post
    If A and B are bounded and non-empty subsets of \mathbb{R}. Prove the following:

    if c > 0, supremum cA = c supremum A.
    Let \alpha=sup(A). Since \alpha is an upper bound for A, c\alpha\geq\\ca for all a\in{A}. Therefore, c\alpha is an upper bound for A. Moreover, if we let \beta be any other upper bound for cA, we have that \frac{\beta}{c} is an upper bound for A. Thus, \alpha\leq\frac{\beta}{c} \Leftrightarrow\\c\alpha\leq\beta. So, c\alpha satisfies the definition of sup(cA).

    That is my stab at it.
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  3. #3
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    Wow, I left out a piece sorry!
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