Null Sequences form Maximal Left and Right Ideal/Lemma 2

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

Let $\CC$ be the ring of Cauchy sequences over $R$

Let $\NN$ 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:
 * $\NN$ is an ideal of $\CC$.

Hence $\NN$ is a left ideal of $\CC$.

It remains to show that $\NN$ is maximal.

Lemma 2.2
The result follows by definition of maximal left ideal.