Null Sequences form Maximal Left and Right Ideal/Lemma 4

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:
 * $\mathcal N \ne \O$

Proof
From Constant Sequence Converges to Constant in Normed Division Ring, the zero $\tuple {0, 0, 0, \dots}$ of $\mathcal C$ to converges $0 \in R$.

Therefore $\tuple {0, 0, 0, \dots} \in \mathcal N$.