Let $(e_{1},\ldots,e_{r})$ be an $e^{-\epsilon/2}$-orthogonal basis of $V$ with respect to $\|\raisebox{1.72218pt}{.}\|$ (cf. Proposition 1.3). As $\|\raisebox{1.72218pt}{.}\|=\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}}$ by Proposition 1.18, we can find $\lambda_{i}\in k^{\times}$ such that $\|e_{i}\|\leq|\lambda_{i}|\leq e^{\epsilon/2}\|e_{i}\|$ for each $i$. We set $\omega_{i}:=\lambda_{i}^{-1}e_{i}$ ($i=1,\ldots,r$) and $\mathscr{V}^{\prime}:=\mathfrak{o}_{k}\omega_{1}+\cdots+\mathfrak{o}_{k}\omega_{r}$. Note that $\omega_{i}\in\mathscr{V}$ for all $i$, that is, $\mathscr{V}^{\prime}$ is a sub-lattice of $\mathscr{V}$ and $\mathscr{V}^{\prime}$ is finitely generated over $\mathfrak{o}_{k}$. For $c_{1},\ldots,c_{r}\in k$, by Proposition 1.14,

	$\displaystyle\|c_{1}e_{1}+\cdots+c_{r}e_{r}\|_{\mathscr{V}^{\prime}}$	$\displaystyle=\|c_{1}\lambda_{1}\omega_{1}+\cdots+c_{r}\lambda_{r}\omega_{r}\|_{\mathscr{V}^{\prime}}=\max\{|c_{1}\lambda_{1}|,\ldots,|c_{r}\lambda_{r}|\}$
		$\displaystyle\leq e^{\epsilon/2}\{|c_{1}|\|e_{1}\|,\ldots,|c_{r}|\|e_{r}\|\}\leq e^{\epsilon}\|c_{1}e_{1}+\cdots+c_{r}e_{r}\|,$

so that we have $\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}^{\prime}}\leq e^{\epsilon}\|\raisebox{1.72218pt}{.}\|$. ∎