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. 

$\phi(x,u)=(e^{-ixu}-1)/u$ so that
$$
|\phi(x,u)|=2\,\frac{|\sin(ux/2)|}{|u|}\le2\,\frac{\min(1,|\tfrac12xu|)}{|u|}=\min\left(\frac{2}{|u|},|x|\right)
$$
which implies the inequality.

The critical step is
$$e^{iy}-1=e^{iy/2}(e^{iy/2}-e^{-iy/2})=2ie^{iy/2}\sin(y/2)$$ and $|ie^{iy/2}|=1$
