$\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert$ 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.

Use the mean value theorem on $f(u)=\cos u$ with the interval $[y,x]$ or $[x,y]$.
$$-1 \leq \frac{\cos x-\cos y}{x-y}=-\sin (c) \leq 1$$
Because for any $c \in \mathbb{R}$ we have $-1 \leq -\sin (c) \leq 1$.
So we have,
$$|\frac{\cos x-\cos y}{x-y}| \leq 1$$
$$|\cos x-\cos y| \leq |x-y|$$
