First we assume that $|\raisebox{1.72218pt}{.}|$ is discrete. We take a positive integer $a$ such that $e^{-\epsilon a/2}\leq|\varpi|$. We also choose $\alpha\in k^{\times}$ such that

$$|\alpha^{-1}|=\min\{|\gamma|\mid\text{$\gamma\in k^{\times}$ and $\|l^{\otimes a}\|_{Y,h^{a}}\leq|\gamma|$}\}.$$

Then, as $\|l^{\otimes a}\|_{Y,h^{a}}\leq|\alpha^{-1}|\leq|\varpi|^{-1}\|l^{\otimes a}\|_{Y,h^{a}}$, we have

$$e^{-a\epsilon/2}\leq|\varpi|\leq\|\alpha l^{\otimes a}\|_{Y,h^{a}}\leq 1.$$

Next we assume that $|\raisebox{1.72218pt}{.}|$ is not discrete. In this case, $|k^{\times}|$ is dense in $\mathbb{R}_{>0}$ by Lemma 1.15, so that we can choose $\beta\in k^{\times}$ such that

$$e^{-\epsilon/2}\leq\|l\|_{Y,h}/|\beta|\leq 1.$$

Thus if we set $\alpha=\beta^{-1}$ and $a=1$, we have the assertion. ∎