Cauchy Sequence Converges Iff Equivalent to Constant Sequence

Theorem

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.

Let $\CC$ be the ring of Cauchy sequences over $R$

Let $\NN$ be the set of null sequences.

Let $\\CC \,\big / \NN$ be the quotient ring of Cauchy sequences of $\CC$ by the maximal ideal $\NN$.

Let $\sequence {x_n} \in \CC$.

Then $\sequence {x_n}$ converges in $\struct {R, \norm {\,\cdot\,} }$ if and only if:

$\exists a \in R: \sequence {x_n} \in \sequence {a, a, a, \dotsc} + \NN$

where $\sequence {a, a, a, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {a, a, a, \dotsc}$.

Proof

By definition, $\sequence {x_n}$ converges to $a \in R$ if and only if $\ds \lim_{n \mathop \to \infty} \norm {x_n - a} = 0$

Then:

\(\ds \lim_{n \mathop \to \infty} \norm {x_n - a}\)

\(=\)

\(\ds 0\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \sequence {x_n - a}\)

\(\in\)

\(\ds \NN\)

Definition of Null Sequence in Normed Division Ring

\(\ds \leadstoandfrom \ \ \)

\(\ds \sequence {x_n} - \sequence {a, a, a, \dotsc}\)

\(\in\)

\(\ds \NN\)

Definition of Addition in Ring of Cauchy Sequences

\(\ds \leadstoandfrom \ \ \)

\(\ds \sequence {x_n}\)

\(\in\)

\(\ds \sequence {a, a, a, \dotsc} + \NN\)

Element in Left Coset iff Product with Inverse in Subgroup

$\blacksquare$