I am studying from Marsden: Elementary Classical Analysis ($2^{\rm{nd}}$ ed.). I am not able to write down the complete proof of the following theorem (Theorem 6.3.1, page 334). The theorem essentially says that:
Let $A\subset \mathbb{R}^n$ be an open set and let $f:A \to \mathbb{R}^m$ be a differentiable function. Then $f$ is locally Lipschitz, i.e. for each $x_0\in A$, there is $M>0$ and $\delta_0>0$ such that $$\|x-x_0\|<\delta_0 \quad \Rightarrow \quad \|f(x)-f(x_0)\|<M\|x-x_0\|.$$
I could not locate the proof anywhere, nor was able to generalize the one-variable proof to multivariable (For example: here, here. This one is slightly different, with an extra condition of $f'$ being continuous.)
The reason I cannot apply these arguments to multivariable case is all these proofs use MVT.
Can someone help with the proof? OR direct me to a reference / book which has a proof?