Equivalent Norms on Rational Numbers

Theorem
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be norms on the rational numbers $\Q$.

Then $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are equivalent :
 * $\exists \alpha \in \R_{\gt 0}: \forall n \in \N: \norm n_1 = \norm n_2^\alpha$