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
By Subring Test it is sufficient to prove:
 * $(1): \quad \tuple {1,1,1,\dots} \in \mathcal {C}$, where $1$ is the unity of $R$.
 * $(2): \quad \forall \sequence {x_n}, \sequence {y_n} \in \mathcal {C}: \sequence {x_n} + \paren {- \sequence {y_n}} \in \mathcal {C}$
 * $(3): \quad \forall \sequence {x_n}, \sequence {y_n} \in \mathcal {C}: \sequence {x_n} \circ \sequence {y_n} \in \mathcal {C}$

$(1): \quad \tuple {1,1,1,\dots} \in \mathcal {C}$
Let $1$ be the zero in $R$.

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

$(2): \quad \forall \sequence {x_n}, \sequence {y_n} \in \mathcal {C}: \sequence {x_n} + \paren {- \sequence {y_n}} \in \mathcal {C}$
By definition of pointwise addition, the sequence $\sequence {x_n} + \paren {- \sequence {y_n}}$ is the sequence $\sequence {x_n - y_n}$.

By Difference Rule for Normed Division Ring Sequences the sequence $\sequence {x_n - y_n}$ is a Cauchy sequence.

$(3): \quad \forall \sequence {x_n}, \sequence {y_n} \in \mathcal {C}: \sequence {x_n} \circ \sequence {y_n} \in \mathcal {C}$
By definition of pointwise product, the sequence $\sequence {x_n} \circ \sequence {y_n}$ is the sequence $\sequence {x_n \circ y_n}$.

By Product Rule for Normed Division Ring Sequences the sequence $\sequence {x_n \circ y_n}$ is a Cauchy sequence.