Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics II / 1 / a / iii / Solution

False. Take the finite sets $A=B=\{-2,-1\}$. Their suprema are both $-1$, whereas the set of pairwise products is $A\cdot B=\{1,2,4\}$. ConsequentlyNegative factors can reverse an inequality, which prevents the proposed argument with separate suprema.