Integers are Dense in P-adic Integers

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Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\Z_p$ be the $p$-adic integers.

Let $d_p$ be the metric induced by the norm $\norm {\,\cdot\,}_p$ restricted to the $p$-adic integers.


The integers $\Z$ are dense in the metric space $\struct{\Z_p, d_p}$.

Corollary

The integers $\Z$ are dense in the closed ball $\map {B^-_1} 0$.


Proof

From Open Ball Characterization of Denseness it is sufficient to show that every open ball of $\struct {\Z_p, d_p}$ contains an element of $\Z$.


Let $x \in \Z_p$ and $\epsilon \in \R_{>0}$.

By definition the open ball $\map {B_\epsilon} x$ is:

$\map {B_\epsilon} x = \set {y \in \Z_p: \norm y_p < \epsilon}$


From Sequence of Powers of Number less than One then:

$\ds \lim_{n \mathop \to \infty} p^{-n} = 0$

Hence there exists $N \in \N$:

$\forall n \ge N: p^{-n} < \epsilon$

Consider the open ball $\map {B_{p^{-N} } } x$.

Since $0 < p^{-n} < \epsilon$ then:

$\map {B_{p^{-N} } } x \subseteq \map {B_\epsilon} x$.


From Integers are Arbitrarily Close to P-adic Integers then:

$\exists \alpha \in \Z: \alpha \in \map {B_{p^{-N} } } x$

Hence $\alpha \in \map {B_\epsilon} x$.


Since $x$ and $\epsilon$ were arbitrary then every open ball of $\struct {\Z_p, d_p}$ contains an element of $\Z$.

From Open Ball Characterization of Denseness then $\Z$ is dense in the metric space $\struct {\Z_p, d_p}$.

$\blacksquare$


Sources

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