Trivial Norm on Division Ring is Non-Archimedean

Theorem
Let $\left({R, +, \circ}\right)$ be a division ring, and denote its ring zero by $0_R$.

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

is non-archimedean:

$\left\Vert{x + y} \right\Vert \leq \max \left\{ {\left\Vert{x}\right\Vert, \left\Vert{y}\right\Vert} \right\}$.

Proof
If $x,y=0_R$, then $\left\Vert{x}\right\Vert,\left\Vert{y}\right\Vert=0$, which means that $\max \left\{ {\left\Vert{x}\right\Vert, \left\Vert{y}\right\Vert} \right\}=0$. We get:

If $x\ne0_R$ or $y\ne0_R$, then $\left\Vert{x}\right\Vert=1$ or $\left\Vert{y}\right\Vert=1$, which means that $\max \left\{ {\left\Vert{x}\right\Vert, \left\Vert{y}\right\Vert} \right\}=1$. We get: