Prove $|e^{i\theta} -1| \leq |\theta|$

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?

Solution 1:

By the fundamental theorem of calculus $$e^{i\theta}-1=\int_0^\theta ie^{it}\mathrm{d}t$$ Hence...

Solution 2:

Think of it that way : You start at point $1+0i$ and move on the unit circle by an angle of theta. This inequality is just saying that going from $1+0i$ to your point $e^{i\theta}$ by a straight line is shorted than going to it by moving along the circle $r\theta$ (where $r=1$ is the radius of the cicle).

Solution 3:

And this is somehow a proof without words:

$z=e^{i\theta}\\ \theta= \overset{\displaystyle\frown}{AB}\\ AH=\sin(\theta)\\ OH=\cos(\theta)\\ HB=1-\cos(\theta) \\ AB=|e^{i\theta}-1| $