Suppose f: [0,inf) -> R is uniformly Lipschitz and the integral from 0 to inf of |f(x)| < inf. Then|f(x)| converges to 0 as x -> inf. Prove this by proving the contrapositive. ie. negate the limit portion and show via the properties given by f that the integral will be infinite.

I'm lost here. For the limit, I think I'd need to choose an epsilon so that the limit does not converge to 0. Then, I somehow use that f is Lipschitz (not totally sure what that means aside from |f(x) - f(y)| < M |x-y|) to show that the integral is infinite (don't totally know how to show that, either). Any help would be greatly appreciated.


marked as duplicate by Martin R, Community Nov 23 '15 at 6:21

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  • $\begingroup$ Is mine the same, since it has a different form of continuity in the question? $\endgroup$ – user269711 Nov 23 '15 at 5:43
  • $\begingroup$ Lipschitz continuity implies uniform continuity, so the other question is slightly more general. $\endgroup$ – Martin R Nov 23 '15 at 5:49
  • $\begingroup$ Ah, thank you very much! $\endgroup$ – user269711 Nov 23 '15 at 6:21