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|[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.