We assume that the absolute value $|\raisebox{1.72218pt}{.}|$ is not discrete. Let $\|\raisebox{1.72218pt}{.}\|$ be a norm of $V$ and $\mathscr{V}:=(V,\|\raisebox{1.72218pt}{.}\|)_{\leq 1}$. For any $\epsilon>0$, there is a sub-lattice $\mathscr{V}^{\prime}$ of $\mathscr{V}$ such that $\mathscr{V}^{\prime}$ is finitely generated over $\mathfrak{o}_{k}$ and $\|\raisebox{1.72218pt}{.}\|\leq\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}^{\prime}}\leq e^{\epsilon}\|\raisebox{1.72218pt}{.}\|$.