Null Sequences form Maximal Left and Right Ideal

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.

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

Then $\mathcal {N}$ is a ring ideal of $\mathcal {C}$ 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 $\mathcal {N} \subseteq \mathcal {C}$.

The proof is completed in these steps:
 * (1): $\mathcal {N}$ is an ideal of $\mathcal {C}$.


 * (2): $\mathcal {N}$ is a maximal left ideal.


 * (3): $\mathcal {N}$ is a maximal right ideal.