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 $\map {B^-_1} 0$.

$\blacksquare$