= Solution
Two norms $N_1,N_2$ are <equivalent norms> when constants $c,C>0$ satisfy
$$
cN_1(v)\leq N_2(v)\leq CN_1(v)
$$
for every $v$.
For $x,y\in\mathbb R^n$, the reverse <triangle inequality> gives
$$
|g(x)-g(y)|
\leq\left\|\sum_{i=1}^n(x_i-y_i)e_i\right\|
\leq\sum_{i=1}^n|x_i-y_i|\,\|e_i\|
\leq C_0\|x-y\|_2
$$
by the <Cauchy-Schwarz inequality>. Hence $g$ is <Lipschitz continuous>, and therefore continuous.
On the compact Euclidean <unit sphere>, $g$ is positive and continuous, so the <extreme value theorem> gives $0<m\leq g\leq M<\infty$. Homogeneity then yields
$$
m\|x\|_2\leq g(x)\leq M\|x\|_2.
$$
Thus every norm on a <finite-dimensional vector space> is equivalent to the <Euclidean norm>; comparing two such bounds proves \b[any two norms on $V$ are Lipschitz equivalent.]
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