Definition:Convergent Sequence/P-adic Numbers

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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 $x \in \Q_p$ 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 $x \in \Q_p$ if and only if:

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


Definition 3

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

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


Definition 4

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

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


Limit of Sequence

Let $\sequence {x_n}$ converge to $x \in \Q_p$


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