Prove that $|\sin(x)-\sin(y)| \leq |x-y|$, for all $x,y \in \mathbb(R)$. The title says pretty much everything.
I have found that $|\sin(x)|\leq |x|$, and i think that this pretty much implies the above mentioned, but im not certain.

notice that $$|\sin(x)-\sin(y)|=|2\sin(\frac{x-y}{2})\cos(\frac{x+y}{2})|\leq|2\sin(\frac{x-y}{2})|$$
