Why does convergence to zero in two norms imply equivalence?

Question

Let $V$ be a normed space with two norms $f$ and $g$. Prove that these statements are equivalent:

$\{z_n\}$ tends to 0 as $n → ∞$ in $f\;\; $ $\forall \{z_n\}\in V$

Answer 1

Assume there is no $\alpha$ such that $\Vert x\Vert_1\le \alpha\Vert x\Vert_2$ for all $x$. So for every $n\in\Bbb N$ there is some $x_n$ such that $\Vert x_n\Vert_1>n\Vert x_n\Vert_2$. Let

$$z_n=\frac{x_n}{\sqrt n \Vert x_n\Vert_2}$$ so $\Vert z_n\Vert_2=\frac1{\sqrt n}\xrightarrow{n\to\infty}0$ wheras $\Vert z_n\Vert_1\ge \sqrt n\xrightarrow{n\to\infty}+\infty$.