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 $a$