Supremum of the product of two sets

Question

Here is a GRE math subject problem:

63. For any nonempty sets $A,B\subseteq \Bbb R$, let $A\cdot B$ be the set defined by $$A\cdot B=\{xy\,:\, x\in A\wedge y\in B\}$$ If $A$ and $B$ are nonempty bounded sets and if $\sup A>\sup B$, then $\sup(A\cdot B)=$

(A) $\quad\sup(A)\sup( B)$

(B) $\quad\sup(A)\inf(B)$

(C) $\quad\max\{\sup(A)\sup(B),\ \inf(A)\inf(B)\}$

(D) $\quad\max\{\sup(A)\sup(B),\ \sup(A)\inf(B)\}$

(E) $\quad\max\{\sup(A)\sup(B),\ \inf(A)\sup(B),\ \inf(A)\inf(B)\}$

The answer is E while I thought it was C.

When could $\inf(A)\sup(B)=\sup(A\cdot B)$ under the condition $\sup(A)>\sup(B)$?

Answer 1

For instance, when $\inf A>0$ and $\sup B<0$. Since all the products will be negative, you want to minimise the absolute values of each factor.

Answer 2

When $\inf B<0$, $|\inf B|\ge \sup B$, $\sup A>0$ and $\sup A\ge |\inf A|$.