Is it true that $\alpha \beta = \sup(F)$? Let $A$,$B$ be subsets of $\mathbb{R}$ that are nonempty and bounded above. By axiom of completeness, we know $\alpha = \sup A$ and $\beta = \sup B$ exists.
Let $$F = \{xy \text{ such that }x \in A \text{ and } y \in B\}$$ 
Is it true that $\alpha \beta = \sup(F)$?

I assume $\alpha = \sup(A)$ and $\beta = \sup(B)$.
HINT
Consider $A = [-1,0]$ and $B = [-3,-2]$. What are $\alpha$, $\beta$ and $\sup\left(\{xy \text{ such that }x \in A \text{ and } y \in B\}\right)$?
