# Thread: Supremum example

1. ## Supremum example

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

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

then I would have $\displaystyle \alpha\beta=-12$, 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?

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

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

then I would have $\displaystyle \alpha\beta=-12$, 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