Trivial Norm on Division Ring is Non-Archimedean

Theorem
Let $\struct {R, +, \circ}$ be a division ring whose ring zero is $0_R$.

Then the trivial norm $\norm {\, \cdot \,}: R \to \R_{\ge 0}$, which is given by:


 * $\norm x = \begin{cases}

0 & : x = 0_R \\ 1 & : \text{ otherwise} \end{cases}$

is non-archimedean:


 * $\norm {x + y} \le \max \set {\norm x, \norm y}$

Proof
Let $x, y = 0_R$.

Then:
 * $\norm x, \norm y = 0$

Therefore:
 * $\max \set {\norm x, \norm y} = 0$.

Hence:

Let $x \ne 0_R$ or $y \ne 0_R$.

Then:
 * $\norm x = 1$ or $\norm y = 1$

Therefore:
 * $\max \set {\norm x, \norm y} = 1$.

Hence: