Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1/Sufficient Condition
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.
Let $y \in \Q_p$
Let $n, m \in Z$, such that $n < m$.
Let there exist $i \in \Z$:
- $(1): \quad 0 \le i \le p^\paren {m - n} - 1$
- $(2): \quad \norm {y - i p^n}_p \le p^{-m}$
Then:
- $\norm y_p \le p^{-n}$
Proof
We have that P-adic Norm satisfies Non-Archimedean Norm Axioms:.
Hence:
\(\ds \norm y_p\) | \(=\) | \(\ds \norm {y - i p^n + i p^n}_p\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {\norm {y - i p^n}_p, \norm {i p^n}_p}\) | Non-Archimedean Norm Axiom $\text N 4$: Ultrametric Inequality |
By assumption:
- $\norm {y - i p^n} \le p^{-m} \le p^{-n}$
and:
\(\ds \norm {i p^n}_p\) | \(=\) | \(\ds \norm i_p \norm {p^n}_p\) | Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity | |||||||||||
\(\ds \) | \(\le\) | \(\ds 1 \cdot p^{-n}\) | As $i \in \Z \subseteq \Z_p$ | |||||||||||
\(\ds \) | \(=\) | \(\ds p^{-n}\) |
Hence:
- $\max \set {\norm {y - i p^n}_p, \norm {i p^n}_p} \le p^{-n}$
So:
- $\norm y_p \le p^{-n}$
$\blacksquare$