Solution
= Solution
The identity map
$$
I:(V,\|\cdot\|_1)\to(V,\|\cdot\|_2)
$$
is a continuous linear bijection. Since both spaces are Banach, the <bounded inverse theorem> makes $I^{-1}$ continuous, giving $\|x\|_1\leq C_2\|x\|_2$.
Completeness is essential. On the space $c_{00}$ of finitely supported sequences, $\|x\|_2\leq\|x\|_1$, but for $x=(1,\ldots,1,0,\ldots)$ the ratio $\|x\|_1/\|x\|_2=\sqrt n$ is unbounded. Both normed spaces are incomplete.