Let $\mathscr{X}=\bigcup_{i=1}^{N}\mathscr{\operatorname{Spec}}(\mathscr{A}_{i})$ be an affine open covering of $\mathscr{X}$ with the following properties:

1. (1)

   $\mathscr{A}_{i}$ is a finitely generated over $\mathfrak{o}_{k}$ for every $i$.

2. (2)

   $\operatorname{Spec}(\mathscr{A}_{i})_{\circ}\not=\emptyset$ for all $i$.

3. (3)

   There is a basis $\omega_{i}$ of $\mathscr{L}$ over $\operatorname{Spec}(\mathscr{A}_{i})$ for every $i$.

We set $l=a_{i}\omega_{i}$ for some $a_{i}\in A_{i}:=\mathscr{A}_{i}\otimes_{\mathfrak{o}_{k}}k$. By our assumption, $|a_{i}|_{x}\leq 1$ for all $x\in\operatorname{Spec}(A_{i})^{\mathrm{an}}_{\mathscr{A}_{i}}$. Therefore, by Theorem 2.1, $a_{i}$ is integral over $\mathscr{A}_{i}$, so that, by the following Lemma 2.3, we can find $s_{i}\in{S}$ such that $s_{i}a_{i}^{n}\in\mathscr{A}_{i}$ for all $n\geq 0$. We set $s=s_{1}\cdots s_{N}$. Then, as $sa_{i}^{n}\in\mathscr{A}_{i}$ for all $n\geq 0$ and $i=1,\ldots,N$, we have the assertion. ∎