Quotient Ring of Cauchy Sequences is Division Ring

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 the quotient ring $\mathcal {C}\,\big / \mathcal {N}$ is a division ring.

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

Proof
By Null Sequences form Maximal Left and Right Ideal then $\mathcal {N}$ is an ideal of the ring $\mathcal {C}$ that is also a maximal left ideal.

By Maximal Left Ideal iff Quotient Ring is Division Ring then the quotient ring $\mathcal {C} \,\big / \mathcal {N}$ is a division ring