P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number

Theorem
Let $p$ be a prime number.

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

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

Then the sequence of partial sums of the series:
 * $\displaystyle \sum_{n \mathop = m}^\infty \dfrac {d_n} {p^{-n}}$

converges to a $p$-adic number in $\struct{\Q_p, \norm{\,\cdot\,}_p}$.