ScalingStacks

Lemma 2.13. Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional ultrametrically normed vector space over $k$. If $\{v_{i}\}_{i\in\{1,\dots,n\}}$ is a finite set of elements of $V$ such that $\{\lVert v_{i}\rVert\}_{i\in\{1,\dots,n\}}$ are disctinct in $\mathbb{R}_{+}$. Then $\lVert\sum_{i\in\{1,\dots,n\}}v_{i}\rVert=\max_{i\in\{1,\dots,n\}}\lVert v_{i}\rVert$.