Fix $l\in H^{0}(X,L)$. If $|l|_{\mathscr{L}}(x)\leq 1$ for all $x\in X^{\mathrm{an}}$, then there is $s\in\mathfrak{o}_{k}\setminus\{0\}$ such that $sl^{\otimes n}\in H^{0}(\mathscr{X},\mathscr{L}^{\otimes n})$ for all $n\geq 0$.