Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Necessary Condition
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Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $n, m \in Z$, such that $n < m$.
Let $y \in \Q_p$.
Let $\norm{y}_p \le p^{-n}$.
Then there exists $i \in \Z$ such that:
- $(1) \quad 0 \le i \le p^\paren {m - n} - 1$
- $(2) \quad \norm {y - i p^n}_p \le p^{-m}$
Proof
We have that P-adic Norm satisfies Non-Archimedean Norm Axioms.
Hence:
\(\ds \norm y_p\) | \(\le\) | \(\ds p^{-n}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds p^n \norm{y}_p\) | \(\le\) | \(\ds 1\) | multiplying both sides by $p^n$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \norm{p^{-n} }_p \norm{y}_p\) | \(\le\) | \(\ds 1\) | Definition of $p$-adic norm | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \norm{p^{-n} y}_p\) | \(\le\) | \(\ds 1\) | Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {B_1^-} {p^{-n}y}\) | \(=\) | \(\ds \map {B_1^-} 0\) | Characterization of Closed Ball in P-adic Numbers |
From Integers are Dense in Unit Ball of P-adic Numbers:
- $\exists \mathop k \in \Z : \norm{p^{-n} y - k}_p \le p^\paren {n - m}$
From Residue Classes form Partition of Integers:
- $\exists \mathop 0 \le i \le p^\paren {m - n} - 1: p^\paren {m - n} \divides k - i$
By definition of the $p$-adic norm:
- $\norm {k - i}_p \le p^\paren {n - m}$
It follows that:
\(\ds \norm {p^{-n} y - i}_p\) | \(\le\) | \(\ds \max \set {\norm {p^{-n} y - k}_p, \norm {i - k}_p}\) | Corollary to P-adic Metric on P-adic Numbers is Non-Archimedean Metric | |||||||||||
\(\ds \) | \(\le\) | \(\ds p^\paren {n - m}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \norm {p^{-n} }_p \norm {y - i p^n}_p\) | \(\le\) | \(\ds p^\paren {n - m}\) | Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds p^n \norm {y - i p^n}_p\) | \(\le\) | \(\ds p^\paren {n - m}\) | Definition of $p$-adic norm | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \norm {y - i p^n}_p\) | \(\le\) | \(\ds p^{-m}\) | dividing both sides by $p^{-n}$ |
$\blacksquare$