Equivalence Class in P-adic Integers Contains Unique Coherent Sequence

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Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences.

Let $\mathbf a$ be an equivalence class in $\Q_p$ such that $\norm{\mathbf a}_p \le 1$.


Then $\mathbf a$ has exactly one representative that is a coherent sequence.


Corollary

Then $\mathbf a$ has exactly one representative that is a $p$-adic expansion of the form:

$\displaystyle \sum_{n \mathop = 0}^\infty d_n p^n$


Proof

Let $\sequence{\beta_n}$ be a sequence representing $\mathbf a$.

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 Corollary of Characterisation of Cauchy Sequence in Non-Archimedean Norm, $\sequence{\alpha_j}$ is a Cauchy sequence.


Lemma 3

$\sequence{\alpha_n}$ and $\sequence{\beta_n}$ are representatives of the same equivalence class in $\Q_p$.


$\Box$


Then $\sequence{\alpha_j}$ is a representative of $\mathbf a$.


Lemma 4

$\sequence{\alpha_j}$ is the only coherent sequence that represents $\mathbf a$.


$\blacksquare$

See also

Sources