Equivalence of Definitions of Convergence in Normed Division Rings

Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $\sequence {x_n}$ be a sequence in $R$.

Definition 1 iff Definition 2
By definition, the metric induced by the norm $\norm {\, \cdot \,}$ is the mapping $d: R \times R \to \R_{\ge 0}$ defined as:


 * $\map d {x, y} = \norm {x - y}$

From Metric Induced by Norm on Normed Division Ring is Metric, $d$ is a metric.

By definition of a convergent sequence in a metric space, $\sequence{x_n}$ converges to $x \in R$ :
 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \map d {x_n, x} < \epsilon$


 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$
 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$

The result follows.

Definition 1 iff Definition 3
Let $x \in R$.

By definition of norm on a division ring, $\norm {\, \cdot \,}$ is a mapping $\norm {\, \cdot \,}:R \to \R_{\ge 0}$.

Then:
 * $\forall n \in \N: \size {\norm{x_n - x} - 0} = \size {\norm{x_n - x}} = \norm{x_n - x}$

By definition of convergence of a real sequence, the real sequence $\sequence {\norm {x_n - x} }$ converges to $0$ in the reals $\R$
 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {\norm{x_n - x} - 0} < \epsilon$


 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \norm{x_n - x} < \epsilon$
 * $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \norm{x_n - x} < \epsilon$

The result follows.

Definition 2 iff Definition 4
By definition, the metric induced by the norm $\norm {\, \cdot \,}$ is the mapping $d: R \times R \to \R_{\ge 0}$ defined as:


 * $\map d {x, y} = \norm {x - y}$

From Metric Induced by Norm on Normed Division Ring is Metric, $d$ is a metric.

By definition, the topolgy induced by the norm $\norm {\, \cdot \,}$ is the topology $\tau$ induced by the metric $d$.

From Metric Induces Topology, $\tau$ is a topology.

From Sequence Converges to Point Relative to Metric iff it Converges Relative to Induced Topology:
 * $\sequence {x_n}$ converges to $x$ in the metric induced by the norm $\norm {\, \cdot \,}$ $\sequence {x_n}$ converges to $x$ in the topology induced by the norm $\norm {\, \cdot \,}$