# 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.