\begin{definition}[Definition:Ring of Cauchy Sequences]
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.
\end{definition}