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 $n \in \N_{\gt 0}$, let $n \cdot 1_R$ denote the sum of $1_R$ with itself $n$-times. That is:
 * $n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{n \, times}$

Then:
 * $\norm {n \cdot 1_R} \le n$.

Proof
Let $n \in \N_{\gt 0}$.

Then: