= Solution
Choose a <basis> of the finite-dimensional vector space and let $|x|_\infty$ be the maximum absolute coordinate. For any norm $N$, the <triangle inequality> gives
$$
N(x)\leq\left(\sum_{i=1}^nN(e_i)\right)|x|_\infty,
$$
so $N$ is <continuous function>[continuous] in the coordinate topology. On the compact set $\{x:|x|_\infty=1\}$, the positive continuous function $N$ attains a positive minimum $m$ and a finite maximum $M$. By <norm>[absolute homogeneity],
$$
m|x|_\infty\leq N(x)\leq M|x|_\infty.
$$
This proves the <equivalence of norms in finite dimensions>: every norm is equivalent to the maximum norm, and <transitive relation>[transitivity] of these inequalities proves that any two norms on a <finite-dimensional vector space> are Lipschitz equivalent.
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