= Compact-sphere proof of finite-dimensional non-Archimedean norm equivalence
{title2=$c\|x\|_0\leq\|x\|\leq C\|x\|_0$}
On a finite-dimensional vector space over $\mathbb Q_p$, the triangle inequality gives an upper bound by a coordinate maximum norm. The coordinate unit sphere is compact. The other norm is continuous and nonzero there, so has a positive minimum. Scaling by powers of $p$ gives the lower bound. The resulting <equivalence of norms in finite dimensions> proves completeness of finite extensions, and applying the comparison to powers of a field element proves equality of two extending multiplicative <absolute values on a field>.
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