Lebl-Basic-Analysis-I · Problem 5.2.18

The Riemann Integral

Mathematics · properties-of-integral

Problem

Prove a version of the so-called Riemann--Lebesgue Lemma (one of several so named): Suppose f:[a,b]→Rf : [a,b] \to \mathbb{R} is continuous and define the sequence {xn}n=1∞\{x_n\}_{n=1}^\infty by xn:=∫abf(t)sin⁡(nt) dt.x_n := \int_a^b f(t) \sin(nt)\,dt. Prove that lim⁡n→∞xn=0\lim_{n\to\infty} x_n = 0.

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