Null Sequences form Maximal Left and Right Ideal

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.

That is:
 * $\NN = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0 }$

Then $\NN$ is a ring ideal of $\CC$ that is a maximal left ideal and a maximal right ideal.

Corollary
Let $\struct {R, \norm {\, \cdot \,} }$ be a valued field.

Proof
By every convergent sequence is a Cauchy sequence then $\NN \subseteq \CC$.

The proof is completed in these steps:
 * $(1): \quad \NN$ is an ideal of $\CC$.


 * $(2): \quad \NN$ is a maximal left ideal.


 * $(3): \quad \NN$ is a maximal right ideal.