From Spivak's Calculus:
Prove that $|\sin x - \sin y| < |x - y|$ for all $x \neq y$. Hint: the same statement, with $<$ replaced by $\leq$, is a straightforward consequence of a well-known theorem.
Now, I might even be able to prove this somehow (?), but I can't seem to figure out what "well-known theorem" the author is alluding to here... any hints?