Properties of Norm on Division Ring/Norm of Integer

Theorem

Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.

Let $\norm {\,\cdot\,}$ be a norm on $R$.

For all $n \in \N_{>0}$, let $n \cdot 1_R$ denote the sum of $1_R$ with itself $n$-times. That is:

$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$

Then:

$\norm {n \cdot 1_R} \le n$.

Let $n \in \N_{>0}$.

Then:

$\ds \norm {n \cdot 1_R}$

$=$

$\ds \norm {1_R + 1_R + \dots + 1_R}$

$\ds $

$\le$

$\ds \underbrace {\norm {1_R} + \norm {1_R} + \dots + \norm {1_R} }_{\text {$n$ times} }$

$\ds $

$=$

$\ds \underbrace {1 + 1 + \dots + 1 }_{\text {$n$ times} }$

$\ds $

$=$

$\ds n$

$\blacksquare$