Embedding Normed Division Ring into Ring of Cauchy Sequences

Theorem

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

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

Let $\phi: R \to \CC$ be the mapping from $R$ to $\CC$ defined as:

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

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

Then $\phi$ is a ring monomorphism.

Proof

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

Let $i: \CC \to R^\N$ be the inclusion mapping of $\CC$ 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 $a \in R$, we have that:

$\map {\paren {i \circ \phi} } a = \map \phi a$

The result follows.

$\blacksquare$