Null Sequences form Maximal Left and Right Ideal/Lemma 2

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:
 * $\mathcal {N}$ is a maximal left ideal.

Proof
By Lemma 1 of Convergent Sequences to Zero are Maximal Ideal then $\mathcal {N}$ is an ideal of $\mathcal {C}$.

Hence $\mathcal {N}$ is an left ideal of $\mathcal {C}$.

It remains to show that $\mathcal {N}$ is maximal.

Lemma 2.2
By the definition of maximal left ideal then the result follows.