 Step-function proof of the Riemann-Lebesgue lemma

For a continuous real function $f$ on $[a,b]$, uniform step approximation on a compact interval gives a finite step function $g={\sum }_j{c}_j{1}_{I_j}$ with small uniform error. Each cosine integral over an interval has absolute value at most $2/|\omega |$. ThereforeFirst 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.