P-adic Norm of p-adic Number is Power of p/Proof 2

Proof
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences.

That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where:
 * $\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$
 * $\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

Then $x$ is a left coset in $\CC \, \big / \NN$.

Let $\sequence{x_n}$ be any Cauchy sequence in $x$.

From Lemma:
 * $\exists v \in \Z: \displaystyle \lim_{n \mathop \to \infty} \norm{x_n}_p = p^{-v}$

By definition of the $p$-adic norm on the $p$-adic numbers as a quotient of Cauchy sequences:
 * $\norm x_p = \displaystyle \lim_{n \mathop \to \infty} \norm{x_n}_p = p^{-v}$