Quotient of Cauchy Sequences is Metric Completion/Lemma 2

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

Let $d$ be the metric induced by $\struct {R, \norm {\, \cdot \,} }$.

Let $\struct {\mathcal {C} \,\big / \mathcal {N}, \norm {\, \cdot \,} }$ be the normed quotient ring of Cauchy sequences over $R$.

Let $d'$ be the metric induced by the norm $\norm {\, \cdot \,}$ on $\mathcal {C} \,\big / \mathcal {N}$.

Let $\struct{\mathcal C / \sim,\tilde d}$ is the metric completion of $\struct {R,d}$.

Then:
 * $\quad d' = \tilde d$

Proof
Let $\sequence{x_n}$ and $\sequence{y_n}$ be cauchy sequences in $\mathcal {C} \,\big / \mathcal {N} = \mathcal C \left[{R}\right] / \sim$ then:

The result follows.