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 $\Z$:

$\forall x, y \in \Z: d^\Z_p \left({x, y}\right) = \left\Vert{x - y}\right\Vert_p$

where $\left\Vert{x - y}\right\Vert_p$ denotes the $p$-adic norm.


Then $d^\Z_p$ is a metric.


Proof

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

$\forall x, y \in \Q: d_p \left({x, y}\right) = \left\Vert{x - y}\right\Vert_p$

forms a metric space $\left({\Q, d_p}\right)$.

The mapping:

$\forall x, y \in \Z: d^\Z_p \left({x, y}\right) = \left\Vert{x - y}\right\Vert_p$

is the restriction of $d_p$ to the integers.

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

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

$\blacksquare$


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