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 $\norm n \gt 1$

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

By Norm of Unity then:
 * $n_0 \gt 1$

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

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

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

By Norm of Negative then:
 * $\forall n \in \N: \norm {-n} = \norm {n} = \size {n}^\alpha =\size {-n}^\alpha$

Hence:
 * $\forall k \in \Z: \norm {k} = \size {k}^\alpha$

By Norm of Inverse then:
 * $\forall k \in \Z: \norm {1/k} = 1/{\norm {k}} = 1/{\size {k}^\alpha} =\size {1/k}^\alpha$

By Norm Axiom (N2) (Multiplicativity) then:
 * $\forall a/b \in \Q: \norm {a/b} = \norm a \norm {1/b} = \size a^\alpha \size {1/b}^\alpha = \size {a/b}^\alpha$

By Definition 5 of Equivalent Norms then $\norm {\, \cdot \,}$ is equivalent to the absolute value $\size {\, \cdot \,}$.