In addition to this, I want to show that if converges, then we have
You know that F(t) is bounded, say for some constant M. Then . Again, when you divide by , that expression will go to 0 as .
To estimate , split it up as . The first of those two integrals is independent of x, so when we divide by g(x) and let , it will go to 0. Thus we need only look at the second of the two integrals, .
Integrating by parts as previously, we get . Use the fact that to write this as . The first two terms on the right side of that equation are bounded, so when we divide by g(x) and let they will go to 0. Thus we need only estimate the size of the integral term.
For that, using the inequality , we get . When we divide by g(x), that gives a result that is essentially less than .
Putting everything together, that should show that .