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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?

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marked as duplicate by Stefan Hansen, robjohn Jan 12 '13 at 14:07

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up vote 10 down vote accepted

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.

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Did it. Thanks! –  Leif Ericson Jan 12 '13 at 11:46
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