Cauchy Sequences form Ring with Unity

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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.


By Constant Rule for Cauchy sequences:

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

Hence:

$\CC \neq \O$


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


By definition of pointwise addition:

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

By Difference Rule for Normed Division Ring Sequences:

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

Hence:

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


By definition of pointwise product:

$\sequence {x_n} \circ \sequence {y_n} = \sequence {x_n \circ y_n}$.

By Product Rule for Normed Division Ring Sequences:

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.

$\blacksquare$


Sources

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