I am asked to prove the mean value inequality for vector valued functions using the following fact:
Let $x,y,z\in \mathbb{R}^{m}$ and $a<b<c$ be real numbers. Then $\frac{z-x}{c-a}$ belongs to the closed line segment joining $\frac{y-x}{b-a}$ and $\frac{z-y}{c-b}$.
I must use this fact to show that if $a,b\in \mathbb{R}^{m}$, $f$ is a differentiable function on a neighbourhood of $[a,b]$ into $\mathbb{R}^{n}$, then there is an $x\in[a,b]$ such that $\|f(b)-f(a)\|\leq \|b-a\|\|f'(x)\|$.
I know how to prove the mean value inequality using the mean value theorem for functions of a single variable and the Cauchy Schwarz inequality (as in Rudin, for example). However, I do not know where the above fact (which is easy to prove) is used. I would be grateful for a hint. Thanks.