Integers are Dense in P-adic Integers/Unit Ball

From ProofWiki
Jump to navigation Jump to search

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

From Closed Balls of P-adic Number:

$\map {B^-_1} 0 = \map {B^-_{p^0}} 0 = 0 + p^0 \Z_p = \Z_p$

From Integers are Dense in P-adic Integers, the integers $\Z$ are dense in the closed ball $\map {B^-_1} 0$.

$\blacksquare$