It is clear that $|D_{v}\varphi(e)|\leq\diam(\tau)\lip_{\tau}(\varphi)$ for all $e,v$. Conversely it is a standard consequence of Rademacher’s theorem that $\lip_{\tau}(\varphi)=\sup_{v\in A}\|\nabla\varphi(v)\|$. For each $v\in A$ we also have $D_{v}\varphi(e)=\langle\nabla\varphi(v),e-v\rangle$. We now claim that there exists $C>0$ such that

$$\|\lambda\|\leq C\sup_{e\in\mathcal{E}(\tau)}|\langle\lambda,v-e\rangle|$$

for all $\lambda\in V^{*}$ and all $v\in\tau$, which will conclude the proof. Indeed the supremum in the right-hand side is a lower semicontinuous function of $(\lambda,v)\in V^{*}\times\tau$. As a consequence it achieves its infimum on the compact set $\{\lambda\in V^{*},\,\|\lambda\|=1\}\times\tau$, and this infimum cannot be zero since $\{v-e,\,e\in\mathcal{E}(\tau)\}$ spans $V$ for each $v\in\tau$. The claim follows by homogeneity. ∎