I know that if $A,B$ are sets of nonnegative numbers, we have that $\sup AB=\sup A \cdot \sup B$, but what happens in the general case, what conditions on $A,B$ do I need for $\sup AB$ to exist?
Sorry for the lack of effort shown, but I've already proved the $A,B\subseteq \Bbb R_+$ case, and I'm a bit lost on this general one.
Hint Try the case when $A$ consists of two negative numbers and $B$ of two negative numbers.
More generally, what happens if all numbers in $A$ hand $B$ are negative?