Completion of Normed Division Ring

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

Then:
 * $\struct {R, \norm {\, \cdot \,} }$ has a normed division ring completion $\struct {R', \norm {\, \cdot \,}' }$

Proof
Let $d$ be the metric induced by $\struct {R, \norm {\, \cdot \,} }$.

Let $\CC$ be the ring of Cauchy sequences over $R$.

Let $\NN = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0_R}$.

Let $\norm {\, \cdot \,}: \CC \, \big / \NN \to \R_{\ge 0}$ be the norm on the quotient ring $\CC \, \big / \NN$ defined by:
 * $\displaystyle \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN } = \lim_{n \mathop \to \infty} \norm{x_n}$

Let $d'$ be the metric induced by $\struct {\CC \, \big / \NN, \norm {\, \cdot \,} }$.

By Quotient Ring of Cauchy Sequences is Normed Division Ring, $\struct {\CC \, \big / \NN, \norm {\, \cdot \,} }$ is a normed division ring.

By Quotient of Cauchy Sequences is Metric Completion, $\struct {\CC \, \big / \NN, d' }$ is the metric completion of $\struct {R, d}$.

Let $\phi: R \to \CC \, \big / \NN$ be the mapping from $R$ to the quotient ring $\CC \,\big / \NN$ defined by:
 * $\quad \quad \quad \forall a \in R: \map \phi a = \tuple {a, a, a, \ldots} + \NN$

where $\tuple {a, a, a, \ldots} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\tuple {a, a, a, \ldots} $.

By Quotient of Cauchy Sequences is Metric Completion, $\map \phi R$ is a dense subset of $\struct {\CC \, \big / \NN, d' }$.

By Embedding Division Ring into Quotient Ring of Cauchy Sequences, $\phi$ is a distance-preserving ring monomorphism.

By the definition of a normed division ring completion, $\struct {\CC \, \big / \NN, \norm {\, \cdot \,} }$ is a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$.