Chapter 4.4, Problem 47ES

One way of proving that $f\left(x\right)\le g\left(x\right)$ for all $x$ in a given interval is to show that $0\le g\left(x\right)-f\left(x\right)$ for all $x$ in the interval; and one way of proving the latter inequality is to show that the absolute minimum value of $g\left(x\right)-f\left(x\right)$ on the interval is nonnegative. Use this idea to prove the inequalities in these exercises.

Prove that $\sin x\le x$ for all $x$ in the interval $\left[0,2\pi \right]$ .