Q:

Doubt in Rudin's Proof:

Once I go through the proof of the below theorem, I could encounter that he used dominated convergence theorem to prove $(f)$, in that how they claim that
$$\frac{e^{-ix(s-t)}-1}{s-t}\leq |x|$$
Kindly explain. 

A:

$$
      \frac{e^{ixu}-1}{u}=\left.\frac{1}{u}e^{ixv}\right|_{v=0}^{u}=\frac{1}{u}\int_{0}^{u}ixe^{ixv}dv. \\
      \left|\frac{e^{ixu}-1}{u}\right| \le \frac{1}{|u|}\int_{0}^{|u|}|x|du=|x|.
$$

