We will use More on Morphisms, Lemma \ref{more-morphisms-lemma-apply-proj-spec} and we will use the notation used and results found More on Morphisms, Section \ref{more-morphisms-section-proj-spec} without further mention; this proof will not make sense without at least understanding the statement of the lemma. Observe that in our case $A = \bigoplus_{m \geq 0} \Gamma(P, \mathcal{L}^{\otimes m})$ is a finite type $k$-algebra all of whose graded parts are finite dimensional $k$-vector spaces, see Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-proper-ample}. \medskip\noindent We may and do think of $s$ as an element $f \in A_1 \subset A$, i.e., a homogeneous element of degree $1$ of $A$. Denote $Y = V(f) \subset X$ the closed subscheme defined by $f$. Then $U \cap Y = (\pi|_U)^{-1}(Q)$ scheme theoretically. Recall the notation $\mathcal{F}_U = \pi^*\mathcal{F}|_U = (\pi|_U)^*\mathcal{F}$. This is a coherent $\mathcal{O}_U$-module. Choose a finite $A$-module $M$ such that $\mathcal{F}_U = \widetilde{M}|_U$ (for existence see Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}). We claim that $H^i_Z(M)$ is annihilated by a power of $f$ for $i \leq s + 1$. \medskip\noindent To prove the claim we will apply Local Cohomology, Proposition \ref{local-cohomology-proposition-annihilator}. Translating into geometry we see that it suffices to prove for $u \in U$, $u \not \in Y$ and $z \in \overline{\{u\}} \cap Z$ that $$ \text{depth}(\mathcal{F}_{U, u}) + \dim(\mathcal{O}_{\overline{\{u\}}, z}) > s + 1 $$ This requires only a small amount of thought. \medskip\noindent Observe that $Z = \Spec(A_0)$ is a finite set of closed points of $X$ because $A_0$ is a finite dimensional $k$-algebra. (The reader who would like $Z$ to be a singleton can replace the finite $k$-algebra $A_0$ by $k$; it won't affect anything else in the proof.) \medskip\noindent The morphism $\pi : L \to P$ and its restriction $\pi|_U : U \to P$ are smooth of relative dimension $1$. Let $u \in U$, $u \not \in Y$ and $z \in \overline{\{u\}} \cap Z$. Let $p = \pi(u) \in P \setminus Q$ be its image. Then either $u$ is a generic point of the fibre of $\pi$ over $p$ or a closed point of the fibre. If $u$ is a generic point of the fibre, then $\text{depth}(\mathcal{F}_{U, u}) = \text{depth}(\mathcal{F}_p)$ and $\dim(\overline{\{u\}}) = \dim(\overline{\{p\}}) + 1$. If $u$ is a closed point of the fibre, then $\text{depth}(\mathcal{F}_{U, u}) = \text{depth}(\mathcal{F}_p) + 1$ and $\dim(\overline{\{u\}}) = \dim(\overline{\{p\}})$. In both cases we have $\dim(\overline{\{u\}}) = \dim(\mathcal{O}_{\overline{\{u\}}, z})$ because every point of $Z$ is closed. Thus the desired inequality follows from the assumption in the statement of the lemma. \medskip\noindent Let $A'$ be the $f$-adic completion of $A$. So $A \to A'$ is flat by Algebra, Lemma \ref{algebra-lemma-completion-flat}. Denote $U' \subset X' = \Spec(A')$ the inverse image of $U$ and similarly for $Y'$ and $Z'$. Let $\mathcal{F}'$ on $U'$ be the pullback of $\mathcal{F}_U$ and let $M' = M \otimes_A A'$. By flat base change for local cohomology (Local Cohomology, Lemma \ref{local-cohomology-lemma-torsion-change-rings}) we have $$ H^i_{Z'}(M') = H^i_Z(M) \otimes_A A' $$ and we find that for $i \leq s + 1$ these are annihilated by a power of $f$. Consider the diagram $$ \xymatrix{ & H^i(U, \mathcal{F}_U) \ar[ld] \ar[d] \ar[r] & \lim H^i(U, \mathcal{F}_U/f^n\mathcal{F}_U) \ar@{=}[d] \\ H^i(U, \mathcal{F}_U) \otimes_A A' \ar@{=}[r] & H^i(U', \mathcal{F}') \ar[r] & \lim H^i(U', \mathcal{F}'/f^n\mathcal{F}') } $$ The lower horizontal arrow is an isomorphism for $i < s$ by Lemma \ref{lemma-alternative-higher} and the torsion property we just proved. The horizontal equal sign is flat base change (Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}) and the vertical equal sign is because $U \cap Y$ and $U' \cap Y'$ as well as their $n$th infinitesimal neighbourhoods are mapped isomorphically onto each other (as we are completing with respect to $f$). \medskip\noindent Applying More on Morphisms, Equation (\ref{more-morphisms-equation-cohomology-torsor}) we have compatible direct sum decompositions $$ \lim H^i(U, \mathcal{F}_U/f^n\mathcal{F}_U) = \lim \left( \bigoplus\nolimits_{m \in \mathbf{Z}} H^i(Q_n, \mathcal{F}_n \otimes \mathcal{L}^{\otimes m}) \right) $$ and $$ H^i(U, \mathcal{F}_U) = \bigoplus\nolimits_{m \in \mathbf{Z}} H^i(P, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) $$ Thus we conclude by Algebra, Lemma \ref{algebra-lemma-daniel-litt}.