Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm

Theorem
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.

Let $x \in \Q$ such that $\norm{x}_p \le 1$.

Then for all $i \in \N$ there exists $\alpha \in \Z$ such that:
 * $\norm{x - \alpha}_p \le p^{-i}$

Proof
Let $i \in \N$.

Let $x = \dfrac a b: a, b \in \Z \text{ and } b \neq 0$.

we can assume that $\dfrac a b$ is in canonical form.

By Valuation Ring of P-adic Norm on Rationals:
 * $\dfrac a b \in \Z_{\paren p} = \set {\dfrac c d \in \Q : p \nmid d}$

So $p \nmid b$.

Since $p \nmid b$, by Prime not Divisor implies Coprime then $p^i \perp b$.

By Integer Combination of Coprime Integers, there exists $m, l \in \Z: mb + lp^i = 1$.

Then:

As $am \in \Z$, the result follows with $\alpha = am$.

Also see

 * Leigh.Samphier/Sandbox/Unique Integers Close to Rationals in Valuation Ring of P-adic Norm