Trivial Norm on Division Ring is Non-Archimedean

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

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


 * $\left\Vert{x}\right\Vert = \begin{cases}

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

is non-archimedean:


 * $\left\Vert{x + y} \right\Vert \le \max \left\{ {\left\Vert{x}\right\Vert, \left\Vert{y}\right\Vert} \right\}$

Proof
Let $x, y = 0_R$.

Then:
 * $\left\Vert{x}\right\Vert, \left\Vert{y}\right\Vert = 0$

Therefore:
 * $\max \left\{ {\left\Vert{x}\right\Vert, \left\Vert{y}\right\Vert} \right\} = 0$.

Hence:

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

Then:
 * $\left\Vert{x}\right\Vert = 1$ or $\left\Vert{y}\right\Vert = 1$

Therefore:
 * $\max \left\{ {\left\Vert{x}\right\Vert, \left\Vert{y}\right\Vert} \right\} = 1$.

Hence: