Partial Sum Congruent to P-adic Integer Modulo Power of p

Theorem

Let $\Z_p$ be the $p$-adic integers for some prime $p$.

Let $a \in \Z_p$.

Let $a = \ds \sum_{j=0}^\infty d_jp^j$ be the $p$-adic expansion of $a$

For all $n \in \N$, let $a_n = \ds \sum_{j=0}^n d_jp^j$ be the n-th partial sum of the $p$-adic expansion of $a$

Then:

$\forall n \in \N : a_n \equiv a \pmod{p^{n+1}\Z_p}$

where $a_n \equiv a \pmod{p^{n+1}\Z_p}$ denotes congruence modulo the ideal $\Z_p$.

We have:

$\ds \forall n \in \N: \, $

$\ds a - a_n$

$=$

$\ds \sum_{j = 0}^\infty d_j p^j - \sum_{j = 0}^n d_j p_j$

Hypothesis

$\ds $

$=$

$\ds \sum_{j = n+1}^\infty d_j p^j$

Removing first $n$ terms from the series

$\ds $

$=$

$\ds p^{n+1} \sum_{j = 0}^\infty d_{j+n+1} p^j$

Extract common factor from the series

$\ds $

$\in$

$\ds p^{k+1} \Z_p$

Definition of P-adic Integer

$\ds \leadsto \ \ $

$\ds \forall k \in \N: \, $

$\ds a$

$\equiv$

$\ds a_k \pmod {p^{k+1}\Z_p}$

$\blacksquare$