  The Riemann-Lebesque Theorem is commonly proved in a few strokes using the
theory of Lebesque integration. Here, the upper bound $2\pi|c_k(f)|\le
S_k(f)-s_k(f)$ 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.
