I know that if $A$ and $B$ are subsets of positive real numbers and bounded respectively, $\rm{sup}(AB)=\rm{sup}(A)\rm{sup}(B)$ holds. However, is this formula still valid when $A$, $B$ consist of nonnegative real numbers?
By looking at the proof for positive cases, I think the answer is yes, but I post the question here for check.
If $A$ is any set of non-negative numbers then supremum of $A$ is same a supremum of $A \setminus \{0\}$ unless $A=\{0\}$. By considering the proof for the cases $A=\{0\}$ and $B=\{0\}$ we see that the answer to your question is YES.