Note that $\mathcal{O}_{Y, y}$ is a Noetherian local ring. Consider the canonical morphism $c : \Spec(\mathcal{O}_{Y, y}) \to Y$, see Schemes, Equation (\ref{schemes-equation-canonical-morphism}). This is a flat morphism as it identifies local rings. Denote momentarily $f' : X' \to \Spec(\mathcal{O}_{Y, y})$ the base change of $f$ to this local ring. We see that $c^*R^pf_*\mathcal{F} = R^pf'_*\mathcal{F}'$ by Lemma \ref{lemma-flat-base-change-cohomology}. Moreover, the fibres $X_y$ and $X'_y$ are identified. Hence we may assume that $Y = \Spec(A)$ is the spectrum of a Noetherian local ring $(A, \mathfrak m, \kappa)$ and $y \in Y$ corresponds to $\mathfrak m$. In this case $R^pf_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y = H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})$ for all $p \geq 0$. Denote $f_y : X_y \to \Spec(\kappa)$ the projection. \medskip\noindent Let $B = \text{Gr}_\mathfrak m(A) = \bigoplus_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1}$. Consider the sheaf $\mathcal{B} = f_y^*\widetilde{B}$ of quasi-coherent graded $\mathcal{O}_{X_y}$-algebras. We will use notation as in Section \ref{section-theorem-formal-functions} with $I$ replaced by $\mathfrak m$. Since $X_y$ is the closed subscheme of $X$ cut out by $\mathfrak m\mathcal{O}_X$ we may think of $\mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F}$ as a coherent $\mathcal{O}_{X_y}$-module, see Lemma \ref{lemma-i-star-equivalence}. Then $\bigoplus_{n \geq 0} \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F}$ is a quasi-coherent graded $\mathcal{B}$-module of finite type because it is generated in degree zero over $\mathcal{B}$ abd because the degree zero part is $\mathcal{F}_y = \mathcal{F}/\mathfrak m \mathcal{F}$ which is a coherent $\mathcal{O}_{X_y}$-module. Hence by Lemma \ref{lemma-graded-finiteness} part (2) we see that $$ H^p(X_y, \mathfrak m^n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d}) = 0 $$ for all $p > 0$, $d \geq d_0$, and $n \geq 0$. By Lemma \ref{lemma-relative-affine-cohomology} this is the same as the statement that $ H^p(X, \mathfrak m^n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0 $ for all $p > 0$, $d \geq d_0$, and $n \geq 0$. \medskip\noindent Consider the short exact sequences $$ 0 \to \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^n \mathcal{F} \to 0 $$ of coherent $\mathcal{O}_X$-modules. Tensoring with $\mathcal{L}^{\otimes d}$ is an exact functor and we obtain short exact sequences $$ 0 \to \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to 0 $$ Using the long exact cohomology sequence and the vanishing above we conclude (using induction) that \begin{enumerate} \item $H^p(X, \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$, $d \geq d_0$, and $n \geq 0$, and \item $H^0(X, \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) \to H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})$ is surjective for all $d \geq d_0$ and $n \geq 1$. \end{enumerate} By the theorem on formal functions (Theorem \ref{theorem-formal-functions}) we find that the $\mathfrak m$-adic completion of $H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})$ is zero for all $d \geq d_0$ and $p > 0$. Since $H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})$ is a finite $A$-module by Lemma \ref{lemma-proper-over-affine-cohomology-finite} it follows from Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}) that $H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})$ is zero for all $d \geq d_0$ and $p > 0$. For $p = 0$ we deduce from Lemma \ref{lemma-ML-cohomology-powers-ideal} part (3) that $H^0(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) \to H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})$ is surjective, which gives the final statement of the lemma.