Let be $f:[0,\infty[ \rightarrow \mathbb{R}$ a decreasing and continuos, $f(x)>0 , \forall x$. If $\displaystyle \int_0^{\infty}f(x)dx$ converges then $\displaystyle \lim_{x \rightarrow \infty} xf(x) = 0.$

How to prove that?

Ok, I know that if $G(x) = \displaystyle \int_0^{x}f(t)dt $ then $ \displaystyle \lim_{x \rightarrow \infty} G(x) = L$, How can I use that for arrive in $xf(x)$??


merged by quid Mar 30 at 20:11

This question was merged with If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim_{x\to\infty} xf(x)=0.$ because it is an exact duplicate of that question.