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:
 * $\mathcal {N} \ne \varnothing$

Proof
The zero $\tuple {0,0,0,\dots}$ of $\mathcal {C}$ converges to $0 \in R$, and therefore $\tuple {0,0,0,\dots} \in \mathcal {N}$.