Assignment:
Prove that if f is integrable on [0, 1], then
limn→∞ ∫10 xn 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.
Provide complete and step by step solution for the question and show calculations and use formulas.