inequality involving complex exponential Is it true that 
$$|e^{ix}-e^{iy}|\leq |x-y|$$ for $x,y\in\mathbb{R}$? I can't figure it out. I tried looking at the series for exponential but it did not help.
Could someone offer a hint? 

Hint: $$\begin{align}|e^{ix}-e^{iy}|^2 &= (\cos x-\cos y)^2 + (\sin x - \sin y)^2 \\ &= 2-2(\sin x \sin y +\cos x \cos y) = 2-2\cos (x-y)\end{align}$$
For the last step, see ProofWiki. Now use the series expansion of $\cos(x-y)$ and compare to $(x-y)^2$.
