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

Theorem
Let $p$ be a prime number.

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

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

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

is a Cauchy sequence in the valued field $\struct{\Q, \norm{\,\cdot\,}_p}$.