P-adic Norm not Complete on Rational Numbers/Proof 3

Proof
By Rational Numbers are Countably Infinite, the set of rational numbers is countably infinite.

By P-adic Numbers are Uncountable, the set of $p$-adic numbers $\Q_p$ is uncountably infinite.

Let $\CC$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

The $p$-adic numbers $\Q_p$ is the quotient ring $\CC \, \big / \NN$ by definition.

By Embedding Division Ring into Quotient Ring of Cauchy Sequences, the mapping $\phi: \Q \to \Q_p$ defined by:
 * $\map \phi r = \sequence {r, r, r, \dotsc} + \NN$

where $\sequence {r, r, r, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {r, r, r, \dotsc}$, is a distance-preserving monomorphism.

By Corollary to Surjection from Natural Numbers iff Countable then $\phi$ is not a surjection.

Hence:
 * $\exists \sequence {x_n} \in \CC: \sequence {x_n} + \NN \not \in \map \phi \Q$

By Cauchy Sequence Converges Iff Equivalent to Constant Sequence then $\sequence {x_n}$ is not convergent in $\struct {\Q, \norm {\,\cdot\,}_p}$.

The result follows.