# Cauchy Sequences form Ring with Unity/Corollary

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## 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 $\tuple {1,1,1,\dots}$.

Let $\mathcal {C} \subset F^{\N}$ be the set of Cauchy sequences on $F$.

Then:

- $\struct {\mathcal {C}, +, \circ}$ is a commutative subring of $F^{\N}$ with unity $\tuple {1,1,1,\dots}$.

## Proof

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

By Restriction of Commutative Operation is Commutative the restriction of $\circ$ to $\mathcal {C}$ is commutative since $\circ$ is commutative on $F^{\N}$

## Sources

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