Open Ball in P-adic Numbers is Closed Ball
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Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$:
- Let $\map {B_\epsilon} a$ denote the open $\epsilon$-ball of $a$
- Let $\map {B^-_\epsilon} a$ denote the closed $\epsilon$-ball of $a$.
Then:
- $\forall n \in \Z : \map {B_{p^{-n} } } a = \map {B^-_{p^{-\paren {n + 1} } } } a$
Proof
Let $n \in \Z$.
Then:
\(\ds x \in \map { B_{p^{-n} } } a\) | \(\leadstoandfrom\) | \(\ds \norm {x - a}_p < p^{-n}\) | Definition of Open Ball in $p$-adic Numbers | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \norm {x - a}_p \le p^{-\paren {n + 1} }\) | $p$-adic Norm of $p$-adic Number is Power of $p$ | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in \map {B^-_{p^{-\paren {n + 1} } } } a\) | Definition of Closed Ball in $p$-adic Numbers |
By set equality:
- $\map {B_{p^{-n} } } a = \map {B^-_{p^{-\paren {n + 1} } } } a$
$\blacksquare$