Integers are Dense in P-adic Integers/Unit Ball
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Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
The integers $\Z$ are dense in the closed ball $\map {B^-_1} 0$.
Proof
We have:
\(\ds \map {B^-_1} 0\) | \(=\) | \(\ds \map {B^-_{p^0} } 0\) | Zeroth Power of Real Number equals One | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 + p^0 \Z_p\) | Closed Balls of P-adic Number | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 + 1 \cdot \Z_p\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0 + \Z_p\) | Definition of Multiplicative Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \Z_p\) | Definition of Additive Identity |
From Integers are Dense in P-adic Integers, the integers $\Z$ are dense in the closed ball $\map {B^-_1} 0$.
$\blacksquare$