Coherent Sequence is Partial Sum of P-adic Expansion

Theorem
Let $p$ be a prime number.

Let $\sequence {\alpha_n}$ be a coherent sequence.

Then there exists a unique $p$-adic expansion of the form:
 * $\ds \sum_{n \mathop = 0}^\infty d_n p^n$

such that:
 * $\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n d_i p^i$

Proof
By definition of a coherent sequence:
 * $\forall n \in \N: 0 \le \alpha_n < p^{n + 1}$

From Zero Padded Basis Representation, for all $n \in \N$ there exists a sequence $\sequence {b_{j, n} }_{0 \le j \le n} :$
 * $(1) \quad \ds \alpha_n = \sum_{j \mathop = 0}^n b_{j, n} p^j$
 * $(2) \quad \forall j \in \closedint 0 n : 0 \le b_{j, n} < p$

Lemma
For all $n \in \N$, let:
 * $d_n = b_{n, n}$

Then:
 * $\forall n \in \N : 0 \le d_n < p$

By definition:
 * $\ds \sum_{n \mathop = 0}^\infty d_n p^n$

is a $p$-adic expansion.

From the lemma:
 * $\forall n \in \N: \alpha_n = \ds \sum_{i \mathop = 0}^n d_i p^i$