Definition:Convergent Sequence/P-adic Numbers

Definition

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\sequence {x_n} $ be a sequence in $\Q_p$.

Definition 1

The sequence $\sequence {x_n}$ converges to the limit $x \in \Q_p$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x}_p < \epsilon$

Definition 2

The sequence $\sequence {x_n}$ converges to the limit $x \in \Q_p$ if and only if:

$\sequence {x_n}$ converges to $x$ in the $p$-adic norm

Definition 3

The sequence $\sequence {x_n}$ converges to the limit $x \in \Q_p$ if and only if:

$\sequence {x_n}$ converges to $x$ in the $p$-adic metric

Definition 4

The sequence $\sequence {x_n}$ converges to the limit $x \in \Q_p$ if and only if:

the real sequence $\sequence {\norm {x_n - x}_p }$ converges to $0$ in the reals $\R$

Also see

• Equivalence of Definitions of Convergent P-adic Sequence

• Definition:P-adic Norm
• P-adic Norm is Norm
• Definition:Convergent Sequence in Normed Division Ring

• Definition:P-adic Metric
• P-adic Metric is Metric
• Definition:Convergent Real Sequence

• Results about convergent $p$-adic sequences can be found here.