TITLE: Prove that $|e^{i\theta_1}-e^{i\theta_2}| \leq |\theta_1 - \theta_2|$
QUESTION [8 upvotes]: I'm trying to prove the inequality
$$|e^{i\theta_1}-e^{i\theta_2}| \leq |\theta_1 - \theta_2|$$
I have tried to use Taylor's formula and got this
$$|e^{i\theta_1}-e^{i\theta_2}| = |(1+i\theta_1 - \frac{\theta_1^2}{2} + \ldots) -(1 +i\theta_2 - \frac{\theta_2^2}{2} + \ldots)| = |i(\theta_1-\theta_2) + \frac{\theta_2^2-\theta_1^2}{2}+\ldots|$$
The first term looks right, but how do I proceed?

REPLY [3 votes]: Think geometrically. We can represent complex exponential on the unit circle. The quantity $|e^{i\theta_1} - e^{i\theta_2}|$ is the distance between the two points on the circle, the blue line. Geometrically $\theta_1,\theta_2$ are the arc lengths and so $|\theta_1 - \theta_2|$ is the length of the arc that bounds the blue line. So we see that $|\theta_1 - \theta_2|$ is longer - since the line is the shortest distance between two points.