Properties of Norm on Division Ring/Norm of Difference

Theorem
Let $\struct {R, +, \circ}$ be a division ring.

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

Let $x, y \in R$.

Then:
 * $\norm {x - y} \le \norm x + \norm y$

Proof
Then:

as desired.