Prove that $\sup(f(x)\cdot g(x)) \leq \sup(f(x))\cdot \sup(g(x))$ for nonnegative $f$, $g$

Question

Possible Duplicate:
The Supremum and Bounded Functions
Suprema proof: prove $\sup(f+g) \le \sup f + \sup g$

I am trying to prove this inequality but I don't seem to have any lead. Any suggestions?

Answer 1

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.