If $\epsilon > 0$, $c \ge 0$, $B(x,\epsilon) \subseteq \Bbb{R}^p$, and $\phi : B(x,\epsilon) \to \Bbb{R}^q$ is a continuously differentiable function with $||D\phi(x)|| \le c$ for all $x \in B(x,\epsilon)$, show that $||\phi(x_1)-\phi(x_2)|| \le c ||x_1 -x_2||$ whenever $||x_1-x_2|| < \epsilon$.
I've been thinking about this problem for quite some time and I haven't made any progress. Also, the hint the book provides doesn't seem very helpful. The hint refers to Proposition 3.1.10(b) as being helpful, which is
If $f [a,b] \to \Bbb{R}$ is a Riemann Integrable function and $|f| \le M$ on $[a,b]$ for some $M>0$, then $$\left| \int_{a}^{b} f \right| \le M(b-a)$$.
I think the hint is very mistaken, since this theorem seems to have virtually no relevance to the problem at hand. How should I approach it?