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

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:

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

$\blacksquare$