Let $A$ and $B$ be two sets from $\mathbb{R}$
How to prove that
$$\sup(AB)=\max\{\sup(A)\sup(B),\sup(A)\inf(B), \sup(B)\inf(A),\inf(A)\inf(B)\} $$
where $AB=\{a\cdot b, a\in A, b\in B\}$ Edit1:
I prove this by distinguishing 4 cases A positive or negative or B positive or negative
But what about if $A=[-1,4]$ and $ B= [-3,6]$ how to do ?