## Theorem

Let $p$ be a prime number.

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

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

Then the sequence of partial sums of the series:

$\ds \sum_{n \mathop = m}^\infty d_n p^n$

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

### Corollary 1

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

Then the sequence of partial sums of the series:

$\ds \sum_{n \mathop = m}^\infty d_n p^n$

represents a $p$-adic number of $\struct {\Q_p,\norm {\,\cdot\,}_p}$.

### Corollary 2

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

Then the sequence of partial sums of the series:

$\ds \sum_{n \mathop = m}^\infty d_n p^n$

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

## Proof

Let $\sequence {s_N}$ be the sequence of partial sums defined by:

$\forall N \in \Z_{\ge m}: s_N = \ds \sum_{n \mathop = m}^N d_n p^n$
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 $N \in \Z_{\ge m}$:

 $\ds s_{N + 1} - s_N$ $=$ $\ds \sum_{n \mathop = m}^{N + 1} d_n p^ n - \sum_{n \mathop = m}^{N} d_n p^n$ Definition of Partial Sum $\ds$ $=$ $\ds d_{N + 1} p^{N + 1}$ $\ds \leadsto \ \$ $\ds s_{N + 1}$ $\equiv$ $\ds s_N \pmod {p^N}$ Definition of Congruence modulo $p^N$

The result follows.

$\blacksquare$