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

## Contents

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

### Corollary

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

Then the sequence of partial sums of the series:

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

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:

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

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 $N \in \Z_{\ge m}$:

\(\displaystyle s_{N+1} - s_N\) | \(=\) | \(\displaystyle \sum_{n \mathop = m}^{N+1} d_n p^ n - \sum_{n \mathop = m}^{N} d_n p^n\) | Definition of partial sum | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle d_{N+1} p^{N+1}\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle s_{N+1}\) | \(\equiv\) | \(\displaystyle s_N \pmod {p^N}\) | Definition of Congruence modulo $p^N$ |

The result follows.

$\blacksquare$

## Sources

- 2007: Svetlana Katok:
*p-adic Analysis Compared with Real*... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$