Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.2

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

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

Then: :$n_0$ is a prime number.

Proof
Aiming for a contradiction, suppose $n_0$ is a composite number.

Let $n_1, n_2 \in \N$ such that $n_1, n_2 < n_0$ and $n_0 = n_1 n_2$.

By the definition of $n_0$ then:
 * $\norm {n_1} = 1$
 * $\norm {n_2} = 1$

By Norm Axiom (N2) (Multiplicativity) then:
 * $\norm {n_0} = \norm {n_1 n_2} = \norm {n_1} \norm {n_2} = 1$

This contradicts the assumption that $\norm {n_0} \lt 1$.

So $n_0$ must be a prime number.