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:
 * $\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$

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

Proof
To prove this theorem, extensive use of the following lemma is made.

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

From Lemma:
 * $\forall n \in \N : \exists \sequence{b_{j,n}}_{0 \le j \le n} :$
 * $\displaystyle \alpha_n = \sum_{j \mathop = 0}^{n} b_{j,n} p^j$
 * $\forall \, 0 \le j \le n : 0 \le b_{j,n} < p$

Lemma 2
For all $n \in \N$, let:
 * $d_n = b_{n,n}$

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

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

is a $p$-adic expansion.

From Lemma 2:
 * $\forall n \in \N: \alpha_n = \displaystyle \sum_{i \mathop = 0}^n d_i p^i$