P-adic Metric on Integers is Metric

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Definition

Let $p \in \N$ be a prime.


Let $d^\Z_p$ be the $p$-adic metric on $\Z$:

$\forall x, y \in \Z: \map {d^\Z_p} {x, y} = \norm {x - y}_p$

where $\norm {x - y}_p$ denotes the $p$-adic norm.


Then $d^\Z_p$ is a metric.


Proof

From $p$-adic Metric is Metric, the $p$-adic metric on $\Q$:

$\forall x, y \in \Q: \map {d_p} {x, y} = \norm {x - y}_p$

forms a metric space $\struct {\Q, d_p}$.

The mapping:

$\forall x, y \in \Z: \map {d^\Z_p} {x, y} = \norm {x - y}_p$

is the restriction of $d_p$ to the integers.

Hence the $p$-adic metric on $\Z$ is a metric subspace $\struct {\Z, d^\Z_p}$ of $\struct {\Q, d_p}$.

The result follows from Subspace of Metric Space is Metric Space.

$\blacksquare$


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