# Help with qual problem on Riemann-Stieltjes Integration

by TopCat
Tags: integration, qual, riemannstieltjes
 P: 58 Let f:[0, ∞) → ℝ be a decreasing function. Assume that |$\int^{\infty}_{0}$f(x)dx| < ∞ Show that $lim_{x→∞}$xf(x) = 0. Attempt: By hypothesis f is integrable for every b>0, as is the function x, so that for every such b, xf(x) is integrable on [0,b]. From here I tried passing to sums by considering a partition P such that U(P,xf(x)) - L(P,xf(x)) < ε, but I'm probably barking up the wrong tree there. I also can't see a way to use integration by parts to generate an xf(x) term since I don't have differentiability or continuity of f. I'm pretty much just lost and need a slight nudge in the right direction.