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Math Help - Supremum example

  1. #1
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    Supremum example

    Let X and Y be nonempty bounded subsets of \mathbb{R}, where \alpha = Sup X and \beta =Sup(Y). Let Z ={  xy: x \in X \mbox{and} \ y \in Y}. show by example that \alpha\beta \ \neq Sup(Z) in general.

    I was thinking of letting Sup(X) = \alpha = 4 and Sup(Y) = \beta=-3

    then I would have \alpha\beta=-12, therefore the Sup(Z) would have to be greater then -12 in order for the statement to be true, thus the inequality holds. Is this a correct assumption?
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  2. #2
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    Quote Originally Posted by lllll View Post
    Let X and Y be nonempty bounded subsets of \mathbb{R}, where \alpha = Sup X and \beta =Sup(Y). Let Z ={  xy: x \in X \mbox{and} \ y \in Y}. show by example that \alpha\beta \ \neq Sup(Z) in general.

    I was thinking of letting Sup(X) = \alpha = 4 and Sup(Y) = \beta=-3

    then I would have \alpha\beta=-12, therefore the Sup(Z) would have to be greater then -12 in order for the statement to be true, thus the inequality holds. Is this a correct assumption?
    X=(-5,2), Y=(-5,2)

    Sup(Z)=25, Sup(X)Sup(Y)=4

    RonL
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