Chapter 4.8, Problem 27ES

(a) Use the Mean-Value Theorem to show that if $f$ is differentiable on an interval, and if $\left|f^{\prime }\left(x\right)\right|\le M$ for all values of $x$ in the interval, then $\left|f\left(x\right)-f\left(y\right)\right|\le M\left|x-y\right|$ for all values of $x$ and $y$ in the interval.

(b) Use the result in part (a) to show that $\left|\sin x-\sin y\right|\le \left|x-y\right|$ for all real values of $x$ and $y$ .