Unique Integer 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:


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


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

Proof
Let $i \in \N$.

From Integer Arbitrarily Close to Rational in Valuation Ring of P-adic Norm:
 * $\exists \mathop {\alpha'} \in \Z: \norm{x - \alpha'}_p \le p^{-i}$

By Integer is Congruent to Integer less than Modulus, then there exists $\alpha \in \Z$:
 * $\alpha \equiv \alpha' \pmod {p^i}$.
 * $0 \le \alpha \le p^i - 1$

Then $\norm {\alpha' - \alpha}_p \le p^{-i}$

Hence:

Now suppose $\beta \in \Z$ satisfies:
 * a. $\quad0 \le \beta \le p^i - 1$
 * b. $\quad\norm { x -\beta}_p \le p^{-i}$

Then:

Hence $p^i \divides \alpha - \beta$, or equivalently, $\alpha \equiv \beta \pmod {p^i}$

By Initial Segment of Natural Numbers forms Complete Residue System then $\alpha = \beta$.

The result follows.

Also see

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