Basic Analysis I — Real Analysis · Problem 5.2.18

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · riemann-lebesgue-lemma

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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