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 $\mathcal {C}$ be the ring of Cauchy sequences over $R$

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

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

Let $d'$ be the metric induced by $\struct {\mathcal {C} \,\big / \mathcal {N}, \norm {\, \cdot \,} }$

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

By Quotient of Cauchy Sequences is Metric Completion then $\struct {\mathcal {C} \,\big / \mathcal {N}, d' }$ is the metric completion of $\struct {R,d}$.

Let $\phi:R \to \mathcal {C} \,\big / \mathcal {N}$ be the mapping from $R$ to the quotient ring $\mathcal {C} \,\big / \mathcal {N}$ defined by:
 * $\quad \quad \quad \forall a \in R: \phi \paren {a} = (a,a,a,\dots) + \mathcal {N}$

where $(a,a,a,\dots) + \mathcal {N}$ is the left coset in $\mathcal {C} \,\big / \mathcal {N}$ that contains the constant sequence $(a,a,a,\dots)$.

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

By the definition of a normed division ring completion then $\struct {\mathcal {C} \,\big / \mathcal {N}, \norm {\, \cdot \,} }$ is a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$