Quotient Ring of Cauchy Sequences is Normed Division Ring/Corollary 1

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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.

For all $\sequence {x_n} \in \mathcal {C}$, let $\eqclass {x_n}{}$ denote the left coset $\sequence {x_n} + \mathcal {N}$


Let $\norm {\, \cdot \,}_1:\mathcal {C} \,\big / \mathcal {N} \to \R_{\ge 0}$ be defined by:

$\displaystyle \forall \eqclass {x_n}{} \in \mathcal {C} \,\big / \mathcal {N}: \norm {\eqclass {x_n}{} }_1 = \lim_{n \to \infty} \norm{x_n}$


Then $\struct {\mathcal {C} \,\big / \mathcal {N}, \norm {\, \cdot \,}_1 }$ is a valued field.

Proof

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

By Corollary to Quotient Ring of Cauchy Sequences is Normed Division Ring then $\mathcal C\,\big / \mathcal N$ is a field.

The result follows.

$\blacksquare$


Sources