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

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

Then:
 * $\struct {\mathcal {C}, +, \circ}$ is a subring of $R^{\N}$ with unity $\tuple {1,1,1,\dots}$.

Proof
The Subring Test used to prove the result.

By Constant Rule for Cauchy sequences:
 * the constant sequence $\tuple {1,1,1,\dots}$ is a Cauchy sequences.

Hence:
 * $\mathcal {C} \neq \empty$.

Let $\sequence {x_n}, \sequence {y_n} \in \mathcal {C}$.

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:
 * $\sequence {x_n} + \paren {- \sequence {y_n}} \in \mathcal {C}$.

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:
 * $\sequence {x_n} \circ \sequence {y_n} \in \mathcal {C}$.

By the Subring Test the result follows.