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 $\Z_p$.

Proof
We have: