Null Sequences form Maximal Left and Right Ideal/Lemma 7

Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $\mathcal {C}$ be the ring of Cauchy sequences over $R$

Let $\mathcal {N}$ be the set of null sequences.

Then:
 * $\mathcal {N} \subsetneq \mathcal {C}$.

Proof
By every convergent sequence is a Cauchy sequence then $\mathcal {N} \subseteq \mathcal {C}$.

From Constant Sequence Converges to Constant in Normed Division Ring, the unity $\tuple {1,1,1,\dots}$ of $\mathcal {C}$ converges to $1 \in R$, and therefore $\tuple {1,1,1,\dots} \in \mathcal {C} \setminus \mathcal {N}$

So $\mathcal {N} \subsetneq \mathcal {C}$.