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

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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.

$\blacksquare$

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