# Cauchy Sequences form Ring with Unity

## Theorem

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

Let $\struct {R^\N, +, \circ}$ be the ring of sequences over $R$ with unity $\tuple {1, 1, 1, \dotsc}$.

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

Then:

$\struct {\CC, +, \circ}$ is a subring of $R^\N$ with unity $\tuple {1, 1, 1, \dotsc}$.

### Corollary

Let $\struct {F, +, \circ, \norm {\, \cdot \,} }$ be a valued field.

Let $\struct {F^\N, +, \circ}$ be the commutative ring of sequences over $F$ with unity $\tuple {1, 1, 1, \dotsc}$.

Let $\CC \subset F^\N$ be the set of Cauchy sequences on $F$.

Then:

$\struct {\CC, +, \circ}$ is a commutative subring of $F^\N$ with unity $\tuple {1, 1, 1, \dotsc}$.

## Proof

The Subring Test used to prove the result.

the constant sequence $\tuple {1, 1, 1, \dotsc}$ is a Cauchy sequences.

Hence:

$\CC \neq \O$

Let $\sequence {x_n}, \sequence {y_n} \in \CC$.

$\sequence {x_n} + \paren {-\sequence {y_n}} = \sequence {x_n - y_n}$.
$\sequence {x_n - y_n}$ is a Cauchy sequence.

Hence:

$\sequence {x_n} + \paren {- \sequence {y_n}} \in CC$.

By definition of pointwise product:

$\sequence {x_n} \circ \sequence {y_n} = \sequence {x_n \circ y_n}$.
the sequence $\sequence {x_n \circ y_n}$ is a Cauchy sequence.

Hence:

$\sequence {x_n} \circ \sequence {y_n} \in \CC$.

By the Subring Test the result follows.

$\blacksquare$