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

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

Proof
By Convergent Sequences to Zero is Maximal Left 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