Consider the canonical map $$ \pi^*\mathcal{E} \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1) $$ and twist down by $1$ to get $$ \pi^*(\mathcal{E})(-1) \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})} $$ This is a surjective map from a locally free rank $n + 1$ sheaf onto the structure sheaf. Hence the corresponding Koszul complex is exact (More on Algebra, Lemma \ref{more-algebra-lemma-homotopy-koszul-abstract}). In other words there is an exact complex $$ 0 \to \pi^*(\wedge^{n + 1}\mathcal{E})(-n - 1) \to \ldots \to \pi^*(\wedge^i\mathcal{E})(-i) \to \ldots \to \pi^*\mathcal{E}(-1) \to \mathcal{O}_{\mathbf{P}(\mathcal{E})} \to 0 $$ We will think of the term $\pi^*(\wedge^i\mathcal{E})(-i)$ as being in degree $-i$. We are going to compute the higher direct images of this acyclic complex using the first spectral sequence of Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. Namely, we see that there is a spectral sequence with terms $$ E_1^{p, q} = R^q\pi_*\left(\pi^*(\wedge^{-p}\mathcal{E})(p)\right) $$ converging to zero! By the projection formula (Cohomology, Lemma \ref{cohomology-lemma-projection-formula}) we have $$ E_1^{p, q} = \wedge^{-p} \mathcal{E} \otimes_{\mathcal{O}_S} R^q\pi_*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(p)\right). $$ Note that locally on $S$ the sheaf $\mathcal{E}$ is trivial, i.e., isomorphic to $\mathcal{O}_S^{\oplus n + 1}$, hence locally on $S$ the morphism $\mathbf{P}(\mathcal{E}) \to S$ can be identified with $\mathbf{P}^n_S \to S$. Hence locally on $S$ we can use the result of Lemmas \ref{lemma-cohomology-projective-space-over-ring}, \ref{lemma-identify-functorially}, or \ref{lemma-cohomology-projective-space-over-base}. It follows that $E_1^{p, q} = 0$ unless $(p, q)$ is $(0, 0)$ or $(-n - 1, n)$. The nonzero terms are \begin{align*} E_1^{0, 0} & = \pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})} = \mathcal{O}_S \\ E_1^{-n - 1, n} & = R^n\pi_*\left(\pi^*(\wedge^{n + 1}\mathcal{E})(-n - 1)\right) = \wedge^{n + 1}\mathcal{E} \otimes_{\mathcal{O}_S} R^n\pi_*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)\right) \end{align*} Hence there can only be one nonzero differential in the spectral sequence namely the map $d_{n + 1}^{-n - 1, n} : E_{n + 1}^{-n - 1, n} \to E_{n + 1}^{0, 0}$ which has to be an isomorphism (because the spectral sequence converges to the $0$ sheaf). Thus $E_1^{p, q} = E_{n + 1}^{p, q}$ and we obtain a canonical isomorphism $$ \wedge^{n + 1}\mathcal{E} \otimes_{\mathcal{O}_S} R^n\pi_*\left(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)\right) = R^n\pi_*\left(\pi^*(\wedge^{n + 1}\mathcal{E})(-n - 1)\right) \xrightarrow{d_{n + 1}^{-n - 1, n}} \mathcal{O}_S $$ Since $\wedge^{n + 1}\mathcal{E}$ is an invertible sheaf, this implies that $R^n\pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1)$ is invertible as well and canonically isomorphic to the inverse of $\wedge^{n + 1}\mathcal{E}$. In other words we have proved the case $d = - n - 1$ of the lemma. \medskip\noindent Working locally on $S$ we see immediately from the computation of cohomology in Lemmas \ref{lemma-cohomology-projective-space-over-ring}, \ref{lemma-identify-functorially}, or \ref{lemma-cohomology-projective-space-over-base} the statements on vanishing of the lemma. Moreover the result on $R^0\pi_*$ is clear as well, since there are canonical maps $\text{Sym}^d(\mathcal{E}) \to \pi_* \mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)$ for all $d$. It remains to show that the description of $R^n\pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)$ is correct for $d < -n - 1$. In order to do this we consider the map $$ \pi^*(\text{Sym}^{-d - n - 1}(\mathcal{E})) \otimes_{\mathcal{O}_{\mathbf{P}(\mathcal{E})}} \mathcal{O}_{\mathbf{P}(\mathcal{E})}(d) \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1) $$ Applying $R^n\pi_*$ and the projection formula (see above) we get a map $$ \text{Sym}^{-d - n - 1}(\mathcal{E}) \otimes_{\mathcal{O}_S} R^n\pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d)) \longrightarrow R^n\pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(-n - 1) = (\wedge^{n + 1}\mathcal{E})^{\otimes -1} $$ (the last equality we have shown above). Again by the local calculations of Lemmas \ref{lemma-cohomology-projective-space-over-ring}, \ref{lemma-identify-functorially}, or \ref{lemma-cohomology-projective-space-over-base} it follows that this map induces a perfect pairing between $R^n\pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(d))$ and $\text{Sym}^{-d - n - 1}(\mathcal{E}) \otimes \wedge^{n + 1}(\mathcal{E})$ as desired.