Partial Sums of P-adic Expansion forms Coherent Sequence

Theorem
Let $p$ be a prime number.

Let $\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$ be a $p$-adic expansion.

Let $\sequence{\alpha_n}$ be the sequence of partial sums; that is:
 * $\forall n \in \N :\alpha_n = \displaystyle \sum_{i \mathop = 0}^n d_i p^i$.

Then $\sequence{\alpha_n}$ is a coherent sequence.

Proof
From the definition of a coherent sequence, it needs to be shown that $\sequence{\alpha_n}$ is a sequence of integers such that:
 * $(1): \quad \forall n \in \N: 0 \le \alpha_n \le p^{n + 1} - 1$
 * $(2): \quad \forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$

That the sequence $\sequence{\alpha_n}$ is a sequence of integers follows immediately from the assumption that the series begins at $n = 0$ and so the terms of each summation are integers.

Partial Sums satisfy (1)
As none of the coefficients $d_i$ in $\displaystyle \sum_{i \mathop = 1}^s d_i p^i$ is (strictly) negative, the summation itself likewise cannot be negative.

Note
The theorem is stated for the special set of $p$-adic expansions where the series index begins at $0 \in \Z$ and not the more general $m \in \Z_{\le 0}$.

From, it is seen that this set of $p$-adic expansions is indeed the $p$-adic integers.

Also see

 * Coherent Sequence is Partial Sum of P-adic Expansion