Let $A, B \subset (0, \infty)$ be non-empty and bounded from above. Construct the set of products:
$P(AB) = \{x \in (0, \infty)| \exists a \in A, \exists b \in B \text{ such that }x = ab\}$.
Show that $\sup\big(P(AB)\big) = \sup(A)\cdot \sup(B)$
My initial idea is to show $\leq$ and $\geq$. I have some vague ideas, but I have a lot of blind spots with regards to inequalities, so I would be very grateful for help and explanations.