P-adic Integer has Unique Coherent Sequence Representative
Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$ such that $\norm a _p \le 1$.
Then $a$ has exactly one representative that is a coherent sequence.
Corollary
Then $a$ has exactly one representative that is a $p$-adic expansion of the form:
- $\ds \sum_{n \mathop = 0}^\infty d_n p^n$
Proof
Let $\sequence{\beta_n}$ be a sequence representing $a$.
That is, $\sequence{\beta_n}$ is a Cauchy sequence in the $p$-adic norm $\norm{\,\cdot\,}_p$ on the rational numbers $\Q$.
By definition of a Cauchy sequence:
- $\forall j \in \N : \exists \mathop {\map M j} \in \N: \forall m, n \in \N: m, n \ge \map M j : \norm {x_n - x_m} < p^{-j}$
For all $j \in \N$, let:
- $\map N j = \max \set{j, \map M j}$
From P-adic Norm of p-adic Number is Power of p,
- $\forall j \in \N : \exists \mathop {\map N j} \ge j : \forall m, n \in \N: m, n \ge \map N j : \norm {\beta_n - \beta_m} \le p^{-\paren{j + 1}}$
Lemma 1
- $\forall j \in \N: \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$
$\Box$
From Unique Integer Close to Rational in Valuation Ring of P-adic Norm:
- for all $j \in \N$ there exists $\alpha_j \in \Z$:
- $(1): \quad \norm{\beta_{\map N j} - \alpha_j}_p \le p^{-\paren{j + 1}}$
- $(2): \quad 0 \le \alpha_j \le p^{j + 1} - 1$
Lemma 2
- $\forall j \in \N: \norm {\alpha_{j + 1} - \alpha_j }_p \le p^{-\paren {j + 1} }$
$\Box$
By definition of the $p$-adic norm,
- $\forall j \in \N : \alpha_{j + 1} \equiv \alpha_j \pmod {p^{j + 1}}$
Then $\sequence{\alpha_j}$ is a coherent sequence by definition.
From Sequence of Consecutive Integers Modulo Power of p is Cauchy in P-adic Norm:
- $\sequence{\alpha_j}$ is a Cauchy sequence.
Lemma 3
- $\sequence {\alpha_n}$ and $\sequence {\beta_n}$ are representatives of the same $p$-adic number in $\Q_p$.
$\Box$
Then $\sequence{\alpha_j}$ is a representative of $a$.
Lemma 4
- $\sequence {\alpha_j}$ is the only coherent sequence that represents $a$.
$\blacksquare$
Also see
- Coherent Sequence Converges to P-adic Integer
- P-adic Integer has Unique P-adic Expansion Representative
- P-adic Number has Unique P-adic Expansion Representative
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$: Lemma $1.30$