= Step-function proof of the Riemann-Lebesgue lemma
{title2=$\int_a^bf(t)\cos(\omega t)\,dt\to0$}
For a continuous real <function> $f$ on $[a,b]$, <uniform step approximation on a compact interval> gives a finite <step function> $g=\sum_jc_j\mathbf1_{I_j}$ with small uniform error. Each cosine integral over an interval has absolute value at most $2/|\omega|$. Therefore
$$
\left|\int_a^bf(t)\cos(\omega t)\,dt\right|
\le(b-a)\|f-g\|_\infty+\frac2{|\omega|}\sum_j|c_j|.
$$
First choose $g$ to make the first term small, then let $|\omega|$ grow with $g$ fixed. This proves the cosine version of the <Riemann-Lebesgue lemma> without differentiating $f$; the sine and complex-exponential versions follow in the same way.
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