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.

Hint 1: Notice that the derivative of $f(x) = \sin(x)$ is bounded and apply the mean value theorem.
Hint 2: Alternatively, use $\sin(x) - \sin(y) = 2\cos\dfrac{x+y}{2}\sin\dfrac{x-y}{2}$.
