Coherent Sequence is Partial Sum of P-adic Expansion

Theorem
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

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

Then there exists a $p$-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
Let:
 * $\displaystyle \alpha_n = \sum_{j \mathop = 0}^{m_n} b_{j,n} p^j$

be $\alpha_n$ written in base $p$.

Lemma 1

 * $\forall n \in \N : m_n \le n$

Let:
 * $d_n = \begin{cases}

b_{n,n} & \text{if }n = m_n \\ 0 & \text{if }m_n < n \end{cases}$

Lemma 2

 * $\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$ is a p-adic expansion

Let:
 * $\displaystyle \beta_n = \sum_{j \mathop = 0}^{n} d_j p^j$

Then:
 * $\sequence {\beta_n}$ converges since p-adic expansion.

Lemma 3

 * $\alpha_n = \beta_n$ forall $n$.


 * $\sequence{\beta_n}$ are coherent sequence converging to some p-adic integer.