Restricted P-adic Metric is Metric
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Theorem
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.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces: Exercise $8$