Homework Help: Help with qual problem on Riemann-Stieltjes Integration

Let f:[0, ∞) → ℝ be a decreasing function. Assume that

|[itex]\int^{\infty}_{0}[/itex]f(x)dx| < ∞

Show that

[itex]lim_{x→∞}[/itex]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.

 

Thanks! that was just what I needed.

 

I thought I was good, but I'm having trouble seeing where the contradiction lies. I expected that maybe I could choose N large so that f(x)<ε, yet x*f(x)>ε, but I can't see how will work. I know that ∫x*f(x) need not converge so there's nothing there. I also looked at the same N large under the same conditions to see if choosing a partition P such that U(P,x*f(x))-L(P,x*f(x))<ε might imply x*f(x)<ε, but that didn't work either. I must be making dunce move and missing something super obvious here. :(

 

Ah! Got it! I was completely missing out on the fact that f decreasing implies x*f(x) < ∫f dx. Kept focusing too hard on other irrelevant things. Thanks!