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? 

One way is to use
$$
|e^{ix} - e^{iy}| = \left|\int_x^ye^{it}\,dt\right|\leq \int_x^y\,dt = y-x,
$$
assuming $y > x$.
