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any idea?

not really,but it kind of looks like a set up for integrating by parts

I guess so..

F'(x)=f(x)
so we can say F'(t)=f(t) right..

that is what i was thinking, yes

maybe not exactly what you want, but you would get something like \(gF-\int f g'\)

\[\lim_{b \rightarrow} \int\limits_{0}^{b }G(t)F'(t)dt=\]

I guess enough to show that \[\int\limits f g'\] is convergent

first part is no problem, since \(\lim_{t\to \infty}g(t)=0\) and \(\int_a^\infty f(t)\) is bounded

because all you know about \(g'\) is that \(g'\leq 0\)

does this come in a section after some theorem or lemma that maybe you are supposed to use?

no, this is homework, but prof gave the hint and said that you should use integration by part..

well then i guess this is the right approach!