Ostrowski's Theorem/Archimedean Norm

Theorem
Let $\norm {\, \cdot \,}$ be a non-trivial Archimedean norm on the rational numbers $\Q$.

Then $\norm {\, \cdot \,}$ is equivalent to the absolute value $\size {\, \cdot \,}$.

Proof
By Characterisation of Non-Archimedean Division Ring Norms then:
 * $\exists n \in \N$ such that $\norm n > 1$

Let $n_0 = \min \set {n \in \N : \norm n > 1}$

By Norm of Unity then:
 * $n_0 > 1$

Let $\alpha = \dfrac {\log \norm {n_0} } {\log n_0}$

Since $n_0, \norm n_0 > 1$ then:
 * $\alpha > 0$

Lemma 1.2
Hence:
 * $\forall n \in \N: \norm n = n^\alpha = \size n^\alpha$

By Equivalent Norms on Rational Numbers then $\norm {\, \cdot \,}$ is equivalent to the absolute value $\size {\, \cdot \,}$.