Coherent Sequence is Partial Sum of P-adic Expansion

Theorem

Let $p$ be a prime number.

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

Then there exists a unique $p$-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:

 $\displaystyle \alpha_0$ $=$ $\displaystyle d_0$ $0 \le d_0 \le p - 1$ $\displaystyle \alpha_1$ $=$ $\displaystyle d_0 + d_1 p$ $0 \le d_1 \le p - 1$ $\displaystyle \alpha_2$ $=$ $\displaystyle d_0 + d_1 p + d_2 p^2$ $0 \le d_2 \le p - 1$ $\displaystyle \alpha_3$ $=$ $\displaystyle d_0 + d_1 p + d_2 p^2 + d_3 p^3$ $0 \le d_3 \le p - 1$ $\displaystyle \ldots$  $\displaystyle$ $0 \le d_n \le p - 1$

Putting this together, we get the $p$-adic expansion:

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

such that:

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

$\blacksquare$

Proof

By definition of a coherent sequence:

$\forall n \in \N: 0 \le \alpha_n < p^{n + 1}$

From Zero Padded Basis Representation, for all $n \in \N$ there exists a sequence $\sequence{b_{j,n}}_{0 \le j \le n} :$

$(1) \quad \displaystyle \alpha_n = \sum_{j \mathop = 0}^{n} b_{j,n} p^j$
$(2) \quad \forall j \in \closedint 0 n : 0 \le b_{j,n} < p$

Lemma

$\forall n \in \N: \alpha_n = \displaystyle \sum_{i \mathop = 0}^n b_{i,i} p^i$

$\Box$

For all $n \in \N$, let:

$d_n = b_{n,n}$

Then:

$\forall n \in \N : 0 \le d_n < p$

By definition:

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

is a $p$-adic expansion.

From the lemma:

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

$\blacksquare$