\begin{definition}[Definition:Coherent Sequence]
Let $p$ be a prime number.
Let $\sequence {\alpha_n}$ be an integer sequence such that:
:$(1): \quad \forall n \in \N: 0 \le \alpha_n \le p^{n + 1} - 1$
:$(2): \quad \forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$
The sequence $\sequence {\alpha_n}$ is said to be a '''coherent sequence'''.
If it is necessary to emphasize the choice of prime $p$ then the sequence $\sequence {\alpha_n}$ is said to be a '''$p$-adically coherent sequence'''.
\end{definition}