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 $\Q_p$.

The sequence $\sequence {x_n}$ converges to $x \in \Q_p$ in the norm $\norm {\, \cdot \,}_p$ :


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

That is:


 * $\sequence {x_n}$ converges to $x$ in the metric induced by the norm $\norm {\, \cdot \,}_p$

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

Then $x$ is a limit of $\sequence {x_n}$ as $n$ tends to infinity which is usually written:
 * $\displaystyle x = \lim_{n \mathop \to \infty} x_n$

Also see

 * Definition:Metric Induced by Norm
 * Metric Induced by Norm is Metric