I am trying to prove this inequality but I don't seem to have any lead. Any suggestions?
marked as duplicate by Stefan Hansen, robjohn♦ Jan 12 '13 at 14:07
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Hint: If you can prove that $f(x)\cdot g(x) \leq \sup(f)\cdot \sup(g)$, for all $x$, what you want will follow because you will have proven that $\sup(f)\cdot \sup(g)$ is an upper bound of $f\cdot g$ and therefore it's greater than the smallest upper bound.