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Prove that if f is integrable on [0, 1], then
lim n !1 Z
x n f(x)dx = 0
Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrability on R , where they define partition, refinement of a partition, upper and lower Riemann sums, (Riemann) integrability, upper and lower integrals in terms of the infimum and supremum of the respective sums, Riemann sums as the limit, properties of the integral, and the Mean Value Theorems for Integrals.
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The solution is comprised of an explanation for using the Riemann integrability to determine the nature of a function. The solution is detailed and well presented.