Equivalence of Definitions of Equivalent Division Ring Norms

Theorem
Let $\struct{R, \norm{\,\cdot\,}_1}$ and $\struct{R, \norm{\,\cdot\,}_2}$ be normed division rings on the same underlying division ring $R$.


 * (1) $\quad \norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ are topologically equivalent norms.


 * (2) $\quad \norm{\,\cdot\,}_1$ and $\norm{\,\cdot\,}_2$ are Cauchy equivalent norms.


 * (3) $\quad \exists \alpha \in \R_{\gt 0}: \forall x \in R: \norm{x}_1 = \norm{x}_2^\alpha$


 * (4) $\quad \forall x \in R: \norm{x}_1 \lt 1 \iff \norm{x}_2 \lt 1$


 * (5) $\quad \forall x \in R: \norm{x}_1 \ge 1 \iff \norm{x}_2 \ge 1$


 * (6) $\quad \forall x \in R: \norm{x}_1 \le 1 \iff \norm{x}_2 \le 1$


 * (7) $\quad \forall x \in R: \norm{x}_1 \gt 1 \iff \norm{x}_2 \gt 1$