Proving an elementary inequality of real vectors related to the p-Laplacian

Question

How would you prove the following inequality? $$ \left|\left|a\right|^{p-2}a-\left|b\right|^{p-2}b\right|\leq C_p\left|a-b\right|^{p-1} $$ where $a,b\in\mathbb{R}^{n}$ and $C_p>0$ is some constant depending only on $p$, $1<p<2$. The inequality is mentioned in http://www.math.ntnu.no/~lqvist/p-laplace.pdf on page 43 but proof is omitted. The same notes have a collection of inequalities at chapter 10, in particular the following inequality for euclidean inner product product is given $$ (\left|a\right|^{p-2}a-\left|b\right|^{p-2}b)\cdot(a-b)\leq C_{p}\left|a-b\right|^{p} $$ but I'm not sure how to deduce the first inequality from this.