Non-Archimedean Division Ring Iff Non-Archimedean Completion

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

Let $\struct {R', \norm {\, \cdot \,}' }$ be a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$

Then:
 * $\norm {\, \cdot \,}$ is non-archimedean $\norm {\, \cdot \,}'$ is non-archimedean.

Proof
By the definition of a normed division ring completion then:
 * $(1): \quad$ there exists a distance-preserving ring monomorphism $\phi: R \to R'$.
 * $(2): \quad \struct {R', \norm {\, \cdot \,}' }$ is a complete metric space.
 * $(3): \quad \phi \sqbrk R$ is a dense subspace in $\struct {R', \norm {\, \cdot \,}' }$.

By Normed Division Ring is Dense Subring of Completion, $\phi \sqbrk R$ is a dense normed division subring of $R'$ and $\phi: R \to \map \phi R$ is an isometric isomorphism.

By Isometrically Isomorphic Non-Archimedean Division Rings, $\struct {R, \norm {\, \cdot \,} }$ is non-archimedean $\struct {\phi \sqbrk R, \norm {\, \cdot \,}' }$ is.

So it remains to show that $\struct {\phi \sqbrk R, \norm {\, \cdot \,}' }$ non-archimedean $\struct {R', \norm {\, \cdot \,}' }$ is.

Necessary Condition
Let $\struct {\phi \sqbrk R, \norm {\, \cdot \,}' }$ be non-archimedean.

Let $x', y' \in R'$.

By the definition of a dense subset:
 * $\map \cl {\phi \sqbrk R} = R'$

By Closure of Subset of Metric Space by Convergent Sequence then:
 * there exists a sequence $\sequence {x_n'} \subseteq \phi \sqbrk R$ that converges to $x'$, that is, $\ds \lim_{n \mathop \to \infty} x_n' = x'$
 * there exists a sequence $\sequence {y_n'} \subseteq \phi \sqbrk R$ that converges to $y'$, that is, $\ds \lim_{n \mathop \to \infty} y_n' = y'$

By Sum Rule for Sequences in Normed Division Ring then:
 * $\ds \lim_{n \mathop \to \infty} x_n' + y_n' = x' + y'$

By modulus of limit then:
 * $\ds \lim_{n \mathop \to \infty} \norm {x_n'}' = \norm {x'}'$
 * $\ds \lim_{n \mathop \to \infty} \norm {y_n'}' = \norm {y'}'$
 * $\ds \lim_{n \mathop \to \infty} \norm {x_n' + y_n'}' = \norm {x' + y'}'$

By Maximum Rule for Real Sequences then:
 * $\ds \lim_{n \mathop \to \infty} \max \set {\norm {x_n'}', \norm {y_n'}' } = \max \set {\norm {x'}', \norm {y'}'}$

Since $\struct {\phi \sqbrk R, \norm {\, \cdot \,}' }$ is non-archimedean then:
 * $\forall n: \norm {x_n' + y_n'}' \le \max \set {\norm {x_n'}', \norm {y_n'}' }$

By Inequality Rule for Real Sequences then:
 * $\ds \norm {x' + y'}' = \lim_{n \mathop \to \infty} \norm {x_n' + y_n'}' \le \lim_{n \mathop \to \infty} \max \set {\norm {x_n'}', \norm {y_n'}' } = \max \set {\norm {x'}', \norm {y'}'}$

The result follows.

Sufficient Condition
Let $\struct {R', \norm {\, \cdot \,}' }$ be non-archimedean.

By Subring of Non-Archimedean Division Ring then $\struct {\phi \sqbrk R, \norm {\, \cdot \,}' }$ is non-archimedean.