Cauchy Sequence Converges Iff Equivalent to Constant Sequence

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

Let $\mathcal {C}$ be the ring of Cauchy sequences over $R$

Let $\mathcal {N}$ be the set of null sequences.

Let $\mathcal {C} \,\big / \mathcal {N}$ be the quotient ring of Cauchy sequences of $\mathcal {C}$ by the maximal ideal $\mathcal {N}$.

Let $\sequence {x_n} \in \mathcal C$.

Then $\sequence {x_n}$ converges in $\struct {R, \norm{\,\cdot\,}}$
 * $\exists a \in R: \sequence {x_n} \in \sequence {a, a, a, \dotsc} + \mathcal N$

where $\sequence {a, a, a, \dotsc} + \mathcal N$ is the left coset in $\mathcal C \, \big / \mathcal N$ that contains the constant sequence $\sequence {a, a, a, \dotsc}$

Proof
By definition, $\sequence {x_n}$ converges to $a \in R$ $\displaystyle \lim_{n \to \infty} \norm {x_n - a} = 0$

Then: