Is the slope of a function bounded by the supremum of the derivative? For any function $f : \mathbb{R}^{m} \to \mathbb{R}^n$, is it always true that for $x$, $y \in \mathbb{R}^{m}$,
$$\frac{\lVert f(x) - f(y) \rVert}{\lVert x-y \rVert} \leq \sup_{z \in \mathbb R^m}{\lVert D_f(z) \rVert} $$
where $D_f(z)$ is the Jacobian of $f$ evaluated at $z$, assuming $D_f(z)$ exists everywhere?
I'd like to use this as a step in a proof, but am not sure how to show this holds without making any assumptions on $x$ and $y$.

Yes, this is a consequence of the mean value theorem in multiple dimensions
