Embedding Normed Division Ring into Ring of Cauchy Sequences

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Theorem

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

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

Let $\phi:R \to \mathcal {C}$ be the mapping from $R$ to the ring of Cauchy sequences $\mathcal {C}$ defined by:

$\quad \quad \quad \forall a \in R: \phi \paren {a} = \tuple {a,a,a,\dots}$

where $\tuple {a,a,a,\dots}$ is the constant sequence.

Then:

$\phi$ is a ring monomorphism.

Proof

By Cauchy Sequences form Ring with Unity the ring of Cauchy sequences is a subring of the ring of sequences over $R$.

Let $i: \mathcal {C} \to R^{\N}$ be the inclusion mapping of the ring of Cauchy sequences into the ring of sequences.

By Embedding Ring into Ring Structure Induced by Ring Operations the composition $i \circ \phi:R \to R^{\N}$ is a ring monomorphism.

Since for all $a \in R$, $\paren {i \circ \phi} \paren {a} = \phi \paren {a}$

The result follows.

$\blacksquare$

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