P-adic Expansion is a Cauchy Sequence in P-adic Norm

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 $\struct{\Q, \norm{\,\cdot\,}_p}$.

Proof
Let $\sequence {s_N}$ be the sequence of partial sums defined by:
 * $\forall N \in \Z_{\ge m}: s_N = \displaystyle \sum_{n \mathop = m}^N d_n p^n$

From Corollary of Characterisation of Cauchy Sequence in Non-Archimedean Norm the sequence $\sequence {s_N}$ is a Cauchy sequence if:
 * $\forall N \in \Z_{\ge m}: s_{N + 1} \equiv s_N \pmod {p^n}$

Now for all $N \in \Z_{\ge m}$:

The result follows.