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 $\alpha \beta = \sup(F)$?

For a counterexample, take $A=B=[-2,-1]$.
