Coherent Sequence Converges to P-adic Integer

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.

Let $\Z_p$ be the $p$-adic integers.

Then the sequence $\sequence {\alpha_n}$ converges to some $x \in \Z_p$.

That is, there exists $x \in \Z_p$ such that:

$\ds \lim_{n \mathop \to \infty} \alpha_n = x$

Proof

From Coherent Sequence is Partial Sum of P-adic Expansion there exists a unique $p$-adic expansion of the form:

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

such that:

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

From P-adic Expansion Converges to P-adic Number:

$\exists x \in \Q_p : \ds \lim_{n \mathop \to \infty} \alpha_n = x$

By definition of a coherent sequence:

$\forall n \in \N: \alpha_n \in \Z \subseteq \Z_p$

By definition, the $p$-adic integers $\Z_p$ is the closed ball $\map {B^-_1} 0$.

From Closed Ball of Non-Archimedean Division Ring is Clopen, $\Z_p$ is closed in $\norm{\,\cdot\,}_p$.

From Subset of Metric Space contains Limits of Sequences iff Closed:

$x \in \Z_p$

The result follows.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$