Coherent Sequence Converges to P-adic Integer

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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:

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

Proof

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

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

such that:

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

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

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

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