Theorem

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

Let $\Z_p$ be the $p$-adic integers for some prime $p$.

Let $\Q$ be the rational numbers.

Then:

$\Z_p \cap \Q = \set{\dfrac a b \in \Q : p \nmid b}$

Proof

Let $\norm{\,\cdot\,}^\Q _p$ denote the $p$-adic norm on the rational numbers.

We have:

 $\ds \Z_p \cap \Q$ $=$ $\ds \set{\dfrac a b \in \Q : \norm {\dfrac a b}_p \le 1}$ Definition of $p$-adic integers $\ds$ $=$ $\ds \set{\dfrac a b \in \Q : \norm{\dfrac a b}^\Q_p \le 1}$ Rational Numbers are Dense Subfield of P-adic Numbers $\ds$ $=$ $\ds \set{\dfrac a b \in \Q : p \nmid b}$ Valuation Ring of P-adic Norm on Rationals

$\blacksquare$