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

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

Then:
 * $\norm {n_0} = n_0^\alpha$
 * $n_0^\alpha > 1$

Lemma 1.2
These two results imply $\left \Vert {n}\right \Vert = n^\alpha$.

By the second property of norms (namely multiplicativity), this result extends to all $q \in \Q$.

Suppose a series $\left\{{x_1, x_2, \ldots}\right\}$ is Cauchy on the [Definition:Euclidean Metric on Real Number Line|Euclidean metric]].

We have $\left \Vert {x_j - x_i}\right \Vert \le \left \vert {x_j - x_i}\right \vert$, and so the series is Cauchy on $\left \Vert {*}\right \Vert $.

Now suppose a series is Cauchy on $\left \Vert {*}\right \Vert$.

Then for any $N$ such that $\forall i, j > N: \log_\alpha \left \vert{x_j - x_i}\right \vert < \epsilon, \left \Vert {x_j - x_i}\right \Vert < \epsilon$, so the series is Cauchy on the Euclidean metric.

Thus, $\left \Vert {*}\right \Vert$ is Cauchy equivalent to the Euclidean metric.