TITLE: inequality involving complex exponential
QUESTION [21 upvotes]: 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?

REPLY [22 votes]: A graphical hint (apparently I need 30 characters... doesn't stackexchange know that a picture is worth a thousand words?).