First we assume that $|\raisebox{1.72218pt}{.}|$ is discrete. We choose a positive integer $a$ such that $|\varpi|^{-1}\leq e^{a\epsilon/2}$. We set $\mathscr{H}:=\{s\in H^{0}(X,L^{\otimes a})\mid\|s\|_{h^{a}}\leq 1\}$. Note that $\mathscr{H}$ is a finitely generated lattice of $H^{0}(X,L^{\otimes a})$ by Proposition 1.17. As $\|\raisebox{1.72218pt}{.}\|_{h^{a}}\leq\|\raisebox{1.72218pt}{.}\|_{\mathscr{H}}\leq|\varpi|^{-1}\|\raisebox{1.72218pt}{.}\|_{h^{a}}$ by Proposition 1.17, we have the assertion.

Next we assume that $|\raisebox{1.72218pt}{.}|$ is not discrete. By Proposition 1.18, there is a lattice $\mathscr{V}$ of $H^{0}(X,L)$ such that $\|\raisebox{1.72218pt}{.}\|_{h}=\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}}$. By Proposition 1.19, there is a finitely generated lattice $\mathscr{H}$ of $H^{0}(X,L)$ such that $\mathscr{H}\subseteq\mathscr{V}$ and $\|\raisebox{1.72218pt}{.}\|_{h}\leq\|\raisebox{1.72218pt}{.}\|_{\mathscr{H}}\leq e^{\epsilon/2}\|\raisebox{1.72218pt}{.}\|_{h}$, as desired. ∎