For any $f=(f_{1},\dots,f_{n})\in(k^{\times})^{n}$, the numbers $\{\lvert f_{i}\rvert\lVert e_{i}\rVert\}_{i\in\{1,\dots,r\}}$ are distinct, otherwise there exist $i,j\in\{1,\dots,n\},i\neq j$ such that

$$\log\lVert e_{i}\rVert-\log\lVert e_{j}\rVert=\log\Big|\frac{f_{i}}{f_{j}}\Big|\in\log\lvert k^{\times}\rvert$$

which contradicts the assumption of $\mathbb{Q}$-independence. Hence

$$\Big\|\sum_{i\in\{1,\dots,n\}}f_{i}e_{i}\Big\|=\max_{0\leq i\leq n}\lVert f_{i}e_{i}\rVert$$

by Lemma 2.13. ∎