Closed Ball is Disjoint Union of Smaller Closed Balls in P-adic Numbers/Lemma 1

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 $n < m$.

Then:

$\forall y \in \Q_p: \norm y_p \le p^{-n}$ if and only if 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

Necessary Condition

Let $y \in \Q_p$.

Let $\norm y_p \le p^{-n}$.

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 $p^{-n}$

$\Box$

Sufficient Condition

Let $y\in \Q_p$.

Let there exist $i \in \Z$ such that:

$(1) \quad 0 \le i \le p^{\paren {m - n}} - 1$

$(2) \quad \norm {y - i p^n}_p \le p^{-m}$

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:

$\norm y_p \le p^{-n}$

$\blacksquare$