Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit

Theorem
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.

Let $\sequence{x_n}$ be a rational sequence.

Then:
 * $\sequence{x_n}$ converges to $a$ $\sequence{x_n}$ is a representative of $a$

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

By definition of the $p$-adic numbers:
 * $\Q_p$ is the quotient ring $\CC \, \big / \NN$

where:
 * $\CC$ is the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.

and
 * $\NN$ is the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.

By definition of the $p$-adic numbers, $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ is the norm on the quotient ring $\Q_p$ defined by:
 * $\ds \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$

From Rational Numbers are Dense Subfield of P-adic Numbers, the mapping $\phi: \Q \to \Q_p$ defined by:
 * $\map \phi r = \tuple {r, r, r, \dotsc} + \NN$
 * where $\tuple {r, r, r, \dotsc}$ is the constant sequence

embeds $\Q$ as a dense subfield of $\Q_p$.

The theorem follows immediately from Leigh.Samphier/Sandbox/Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit.