Cauchy Sequences form Ring with Unity/Corollary

Corollary to Cauchy Sequences form Ring with Unity

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 field $F$ is a commutative ring by definition.

From Structure Induced by Commutative Ring Operations is Commutative Ring, the ring of sequences over $F$ is a commutative ring.

Hence $\circ$ is commutative on $F^\N$.

By Cauchy Sequences form Ring with Unity, $\struct {\CC, +, \circ}$ is a subring of $F^\N$.

Hence by Restriction of Commutative Operation is Commutative the restriction of $\circ$ to $\CC$ is commutative.

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions