Using Lagrange Mean Value Theorem prove that for every
a,b are real numbers on has |sina-sinb|<or=|a-b|
Let $\displaystyle a,b\in\mathbb{R}$. We know by the MVT that $\displaystyle \frac{\sin(a)-\sin(b)}{a-b}=f'(c)\implies \left|\sin(a)-\sin(b)\right|\leqslant |f'(c)||a-b|$ for some $\displaystyle c\in(a,b)$. But, $\displaystyle f'(c)=\cos(c)\leqslant1 $ so that $\displaystyle \left|\sin(a)-\sin(b)\right|=\left|\cos(c)\right||a-b|\leqslant |a-b|$