Coherent Sequence is Partial Sum of P-adic Expansion/Informal Proof

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:
 * $\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$

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

Informal Proof
Consider the $\displaystyle \alpha_n$ written in base $p$.

To reduce an integer modulo $p^n$, it is simply a matter of stripping off all but the last $n$ digits.

So the coherence condition:
 * $\alpha_{n + 1} \equiv \alpha_n \mod{p^{n+1}}$

means that the last $n + 1$ digits of both integers are the same.

So the sequence $\sequence{\alpha_n}$ can be written:

Putting this together, we get the $p$-adic expansion:
 * $\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$

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