# Definition:Ring of Cauchy Sequences

## Definition

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

Let $\struct {R^\N, +', \circ'}$ be the ring of sequences over $R$.

Let $\CC$ be the set of Cauchy sequences on $R$.

The **ring of Cauchy sequences over $R$** is the subring $\struct {\CC, +', \circ'}$ of $R^\N$ with unity.

The (pointwise) ring operations on the **ring of Cauchy sequences over $R$** are defined as:

- $\forall \sequence {x_n}, \sequence {y_n} \in R^\N: \sequence {x_n} +' \sequence {y_n} = \sequence {x_n + y_n}$
- $\forall \sequence {x_n}, \sequence {y_n} \in R^\N: \sequence {x_n} \circ' \sequence {y_n} = \sequence {x_n \circ y_n}$

The zero of the **ring of Cauchy sequences** is the sequence $\tuple {0, 0, 0, \dots}$, where $0$ is the zero in $R$.

The unity of the **ring of Cauchy sequences** is the sequence $\tuple {1, 1, 1, \dots}$, where $1$ is the unity of $R$.

By Corollary to Cauchy Sequences form Ring with Unity, if $R$ is a valued field then the **ring of Cauchy sequences over $R$** is a commutative ring with unity.

## Also denoted as

It is usual to use the same symbols for the induced pointwise operations on the **ring of Cauchy sequences over $R$** as for the operations that induce them.

## Also see

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*... (previous) ... (next): $\S 3.2$: Completions: Definition $3.2.4$