Difference of Consecutive terms of Coherent Sequence

Theorem
Let $p$ be a prime number.

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

Then:
 * for all $n \in \N_{>0}$ there exists $c_n \in \N$ such that:
 * $0 \le c_n < p$
 * $\alpha_n - \alpha_{n - 1}  = c_n p^n$