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

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

Without loss of generality 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 $m, l \in \Z: m b + l p^i = 1$.

Then:

\(\ds \norm {\dfrac a b - am}_p\) \(=\) \(\ds \norm {\dfrac a b}_p \norm {1 - a m b}_p\) Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity
\(\ds \) \(\le\) \(\ds \norm {1 - m b}_p\) As $\norm {\dfrac a b}_p \le 1$
\(\ds \) \(=\) \(\ds \norm {l p^i}_p\) As $1 - m b = l p^i$
\(\ds \) \(\le\) \(\ds p^{-i}\) Definition of $p$-adic norm on integers

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

$\blacksquare$


Also see


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