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 a unique $\alpha \in \Z$ such that:


 * $(1): \quad \norm{x - \alpha}_p \le p^{-i}$


 * $(2): \quad 0 \le \alpha \le p^i - 1$