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 $\struct {R, \norm {\,\cdot\,} }$ :
 * $\exists a \in R: \sequence {x_n} \in \sequence {a, a, a, \dotsc} + \NN$

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 $a \in R$ $\ds \lim_{n \mathop \to \infty} \norm {x_n - a} = 0$

Then: