Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.1

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

Then: :$\exists n \in \N: 0 \lt \norm n \lt 1$.

Proof
Since $\norm {\, \cdot \,}$ is a non-trivial then:
 * $\exists \dfrac a b \in \Q : 0 < \norm{\dfrac a b} \mbox{ and }\norm{\dfrac a b} \neq 1$

By Norm of Inverse then:
 * $\norm{\dfrac a b} \gt 1 \implies \norm{\dfrac b a} \lt 1$

Hence either $\norm{\dfrac a b} \lt 1$ or $\norm{\dfrac b a} \lt 1$

assume $\norm{\dfrac a b} \lt 1$

By Norm of Quotient then:
 * $\dfrac {\norm a} {\norm b} \lt 1$

Hence:
 * ${\norm a} \lt \norm b$

Let $n = \size a$ and $m = \size b$ where $\size{\,\cdot\,}$ is the absolute value on $\Q$.

Then $n, m \in \N$

By Norm of Negative then:
 * $\norm n = \norm a$
 * $\norm m = \norm b$

Hence:
 * $\norm n \lt \norm m$

Since $\norm {\, \cdot \,}$ is non-Archimedean then:
 * $\norm m \le 1$

Hence:
 * $\norm n \lt \norm m \le 1$