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Thread: Simple supremum question

  1. #1
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    Simple supremum question

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

    $\displaystyle cA =$ {$\displaystyle ca: a \in A$}.

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

    [SOLVED]
    Last edited by cgiulz; Sep 29th 2009 at 01:15 PM.
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  2. #2
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by cgiulz View Post
    If $\displaystyle A$ and $\displaystyle B$ are bounded and non-empty subsets of $\displaystyle \mathbb{R}$. Prove the following:

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

    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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