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.