I am reading Stein and Shakarchi's Complex Analysis, and I encountered a problem at page 45 while reading the proof of $\int_{0}^{\infty}\frac{1-\cos{x}}{x^2}dx=\frac{\pi}{2}$.
The proof says that $f(z)=\frac{1-e^{iz}}{z^2}=\frac{-iz}{z^2}+E(z)$, with $E(z)$ a bounded function as $z$ tend to $0$. I would like to know how this is done.