# Supremum example

• September 29th 2008, 09:40 PM
lllll
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?
• September 29th 2008, 11:22 PM
CaptainBlack
Quote:

Originally Posted by lllll
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