Null Sequences form Maximal Left and Right Ideal/Lemma 1

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 an ideal of $\mathcal C$.

Proof
The Test for Ideal is applied to prove the result.

Lemma 1.3
By Test for Ideal then the result follows.