P-adic Norm of p-adic Number is Power of p/Proof 1

Proof
By definition of the $p$-adic numbers $\Q_p$, the rational numbers $\Q$ are dense in $\Q_p$.

By the definition of a dense subset then $\map \cl \Q = \Q_p$.

By Closure of Subset of Metric Space by Convergent Sequence then:
 * there exists a sequence $\sequence {x_n} \subseteq \Q$ that converges to $x$.

That is:
 * $\displaystyle \lim_{n \mathop \to \infty} x_n = x$

From Modulus of Limit:
 * $\displaystyle \lim_{n \mathop \to \infty} \norm{x_n}_p = \norm x_p$

By Convergent Sequence in Normed Division Ring is Cauchy Sequence, $\sequence {x_n}$ is a Cauchy sequence in $\struct {\Q_p, \norm {\,\cdot\,}_p}$.

From Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers, $\sequence {x_n}$ is a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$.

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