2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152422The Riemann-Lebesque Theorem is commonly proved in a few strokes using the theory of Lebesque integration. Here, the upper bound $2π|c_k(f)|\le S_k(f)-s_k(f)$ for the Fourier coefficients $c_k$ is proved in terms of majoring and minoring Riemann sums $S_k(f)$ and $s_f(k)$, respectively, for Riemann integrable functions $f(x)$. This proof has been used in a course on methods of applied mathematics.LaTex, 2 pages, to appear in the Classroom Notes section in SIAM ReviewFunctional Analysistext