We have a function $f(x)$ defined on $ [0, +\infty)$.
It is known that $|f'(x)|<2$.
Prove that if $\int_0^{\infty} f(x)\,dx$ converges then $\lim_{x \to +\infty} f(x)=0$.
I don't really know how to use the fact about derivative, I would like to apply LHopital's rule in order to jump from integral to function and then to derivative, but it is not suitable here.
Hint. Show that $|f'(x)|<2$ implies that $f$ is uniformly continuous in $[0, +\infty)$. Then take a look here: $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$