Quotient Ring of Cauchy Sequences is Division Ring/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. Then the quotient ring $\mathcal C\,\big / \mathcal N$ is a field.

Proof
By Quotient Ring of Cauchy Sequences is Division Ring then $\mathcal {C}\,\big / \mathcal {N}$ is a division ring.

By Corollary to Cauchy Sequences form Ring with Unity then $\mathcal C$ is a commutative ring with unity.

By Quotient Ring of Commutative Ring is Commutative then $\mathcal C\,\big / \mathcal N$ is a commutative division ring, that is, a field.