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

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 $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion.

Then:
 * $\ds \sum_{n \mathop = m}^\infty d_n p^n$ converges to $a$ $\ds \sum_{n \mathop = m}^\infty d_n p^n$ is a representative of $a$

Proof
By definition of a $p$-adic expansion:
 * $\ds \sum_{n \mathop = m}^\infty d_n p^n$ is a rational sequence.

The theorem follows from Leigh.Samphier/Sandbox/Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit.