# Math Help - Proof using Lagrange M Mean Value Theorem

1. ## Proof using Lagrange M Mean Value Theorem

Using Lagrange Mean Value Theorem prove that for every
a,b are real numbers on has |sina-sinb|<or=|a-b|

2. Originally Posted by amm345
Using Lagrange Mean Value Theorem prove that for every
a,b are real numbers on has |sina-sinb|<or=|a-b|
Let $a,b\in\mathbb{R}$. We know by the MVT that $\frac{\sin(a)-\sin(b)}{a-b}=f'(c)\implies \left|\sin(a)-\sin(b)\right|\leqslant |f'(c)||a-b|$ for some $c\in(a,b)$. But, $f'(c)=\cos(c)\leqslant1$ so that $\left|\sin(a)-\sin(b)\right|=\left|\cos(c)\right||a-b|\leqslant |a-b|$