Rational Numbers are Dense Subfield of P-adic Numbers

Theorem
Let $p$ be a prime number.

Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$.

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 $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\phi: \Q \to \Q_p$ be the mapping 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 $\sequence {r, r, r, \dotsc}$.

Then:
 * $\Q$ is isometrically isomorphic to $\map \phi \Q$ which is a dense subfield of $\Q_p$.

Proof
By Completion of Normed Division Ring and Normed Division Ring is Field iff Completion is Field, the valued field $\struct {\Q_p, \norm {\, \cdot \,}_p}$ is the completion of $\struct {\Q, \norm {\,\cdot\,}_p}$.

By Embedding Division Ring into Quotient Ring of Cauchy Sequences, the mapping $\phi: \Q \to \Q_p$ is a distance-preserving monomorphism.

By Normed Division Ring is Dense Subring of Completion, $\Q$ is isometrically isomorphic to $\map \phi \Q$ which is a dense subfield of $\Q_p$.