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 Null Sequences form Maximal Left and Right Ideal then $\mathcal N$ is an ideal of $\mathcal C$.

Hence $\mathcal N$ is a 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.