Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Necessary Condition

Theorem
If $\sequence{\map \phi {x_n}}$ converges to $y$ then $\sequence{x_n} \in y$

Lemma 1
Let $\sequence{\map \phi {x_n}}$ converge to $y$.

From Convergent Sequence in Normed Division Ring is Cauchy Sequence:
 * $\sequence{\map \phi {x_n}}$ is a Cauchy Sequence

From Lemma 1:
 * $\sequence{x_n}$ is a Cauchy Sequence

Let $y'$ be the left coset that contains $\sequence{x_n}$.

From sufficient condition:
 * $\ds \lim_{n \mathop \to \infty} \norm{\map \phi {x_n} }_Q = y'$

From Convergent Sequence in Metric Space has Unique Limit:
 * $y = y'$