Null Sequences form Maximal Left and Right Ideal/Corollary 1

Theorem
Let $\struct {F, \norm {\, \cdot \,} }$ be a valued field.

Let $\mathcal C$ be the ring of Cauchy sequences over $F$.

Let $\mathcal N$ be the set of null sequences in $F$.

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

Then $\mathcal N$ is a maximal ring ideal of $\mathcal C$.

Proof
By Null Sequences form Maximal Left and Right Ideal then $\mathcal N$ is a maximal left ideal of $\mathcal C$.

A field is by definition a commutative ring.

In a commutative ring, a maximal left ideal is by definition a maximal ideal.