$\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert$ Prove that $$\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert.$$
My try : $ f(x) =\cos x$ and use the mean value theorem.

since $\left| \sin { x }  \right| \le \left| x \right| $
$$|\cos  x-\cos  y|≤\left| 2\sin { \frac { y-x }{ 2 } \sin { \frac { x+y }{ 2 }  }  }  \right| \le 2\left| \sin { \frac { x-y }{ 2 }  }  \right| \left| \sin { \frac { x+y }{ 2 }  }  \right| =2\left| \frac { x-y }{ 2 }  \right| \cdot 1=\left| x-y \right| $$
