Denote by $ \sup f $ the supremum of the set of images of a function. I proved that if the values are positive, then $$ \sup \left( f \right)\sup \left( g \right) > \sup \left( {fg} \right). $$ When does equality hold?
When is $\sup(f) \sup (g) = \sup(fg)$?
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real-analysis
supremum-and-infimum
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2Could you fix the problem? Equality very seldom holds, since you need, intuitively, for $f$ and $g$ to approach their suprema "in the same place". That said, you cannot possibly have proven that the strict inequality always holds, since equality holds if $f=g$. – 2011-08-15
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0@Yuval: I don't know if that is an accurate rendering of whathever it was that Daniel was in the middle of writing (and which you simply chopped off). Perhaps you might have waited a bit to see what the OP was trying to say in the sentence that got truncated? – 2011-08-15