= Solution
Two <norms> $N_1,N_2$ on the same vector space are <equivalent norms>, or Lipschitz equivalent, if there are constants $0<c\leq C<\infty$ such that $cN_1(v)\leq N_2(v)\leq CN_1(v)$ for every $v$. The constants are uniform over the whole space.
Let $N$ be any <norm> on $\mathbb R^n$. The <triangle inequality> and the <Cauchy-Schwarz inequality> give
$$
N(x)\leq\sum_i|x_i|N(e_i)\leq\left(\sum_iN(e_i)^2\right)^{1/2}\|x\|_2=C\|x\|_2.
$$
The reverse triangle inequality then gives $|N(x)-N(y)|\leq C\|x-y\|_2$, so $N$ is Euclidean-continuous; this continuity was derived, rather than assumed. On the compact Euclidean unit sphere, $N$ attains a strictly positive minimum $c$, because it vanishes only at zero. Homogeneity yields
$$
\boxed{c\|x\|_2\leq N(x)\leq C\|x\|_2.}
$$
Finally a <linear map> $A:\mathbb R^n\to\mathbb R^m$ satisfies $\|Ax\|_2\leq(\sum_{i,j}|a_{ij}|^2)^{1/2}\|x\|_2$, again by <Cauchy-Schwarz inequality>. The equivalence just proved transfers this bound to any norms on the source and target. Hence \b[every such linear map is continuous].
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