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

Proof
Let:
 * $\displaystyle \alpha_n = \sum_{j \mathop = 0}^{m_n} b_{j,n} p^j$

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