P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 1
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Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Let $x \in \Z_p$.
Let $\sequence{\alpha_n}$ be an integer sequence:
- $(1): \quad 0 \le \alpha_n \le p^{n + 1} - 1$
- $(2): \quad \norm {x -\alpha_n}_p \le p^{-\paren{n + 1}}$
Then:
- $\forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$
Proof
For any $n \in \N$ then:
\(\ds \norm {\alpha_{n + 1} - \alpha_n }_p\) | \(=\) | \(\ds \norm {\paren {\alpha_{n + 1} - x} + \paren {x - \alpha_n } }_p\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {\norm {\alpha_{n + 1} - x}_p, \: \norm {x - \alpha_n }_p }\) | Non-Archimedean Norm Axiom $\text N 4$: Ultrametric Inequality | |||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {\norm {x - \alpha_{n + 1} }_p, \: \norm {x - \alpha_n }_p}\) | Norm of negative | |||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {p^{-\paren{n + 2} } , p^{-\paren{n + 1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p^{-\paren{n + 1} }\) | Since $p^{-n - 1} < p^{-n}$ |
Hence:
- $p^{n + 1} \divides \paren {\alpha_{n + 1} - \alpha_n} $
or equivalently:
- $\alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1} }$
The result follows.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.4 \ \text {(ii)}$