TITLE: Prove $|e^{i\theta} -1| \leq |\theta|$
QUESTION [27 upvotes]: Could you help me to prove 
$$
|e^{i\theta} -1| \leq |\theta|
$$
I am studying the proof of differentiability of Fourier Series, and my book used this lemma. How does it work?

REPLY [3 votes]: For real $\theta$ we have $\;|e^{i\theta}-1|^2=\;|\cos \theta +i\sin \theta-1|^2=$ $=(\cos \theta-1)^2+(\sin \theta)^2=1+\cos^2 \theta+\sin^2 \theta-2\cos \theta=$ $=2-2\cos \theta=4((1-\cos \theta)/2)=4\sin^2(\theta /2)\leq 4(\theta$ $ /2)^2=\theta^2.$