P-adic Integer has Unique Coherent Sequence Representative/Lemma 3

Theorem

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rational numbers $\Q$.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\sequence{\beta_n}$ be a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$ such that:

$\forall j \in \N : \exists \mathop {\map N j} \ge j : \forall m, n \in \N: m, n \ge \map N j : \norm {\beta_n - \beta_m} \le p^{-\paren {j + 1} }$

Let $\sequence{\alpha_n}$ be a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$ such that:

$\forall j \in \N: \norm {\alpha_j - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$

Then:

$\sequence {\alpha_n}$ and $\sequence {\beta_n}$ are representatives of the same $p$-adic number in $\Q_p$.

Let $\epsilon \in \R_{> 0}$.

Then there exists $j \in \N$ such that $p^{-j} < \epsilon$.

Then for all $i \ge \map N j$:

$\ds \norm {\alpha_i - \beta_i}_p$

$=$

$\ds \norm {\alpha_i - \beta_{\map N i} + \beta_{\map N i} - \beta_i}_p$

$\ds $

$\le$

$\ds \max \set {\norm {\alpha_i - \beta_{\map N i} }_p, \norm {\beta_{\map N i} - \beta_i}_p}$

$\ds $

$\le$

$\ds \max \set {\norm {p^{-\paren{i + 1} } }_p, \norm {\beta_{\map N i} - \beta_i}_p}$

$\ds $

$\le$

$\ds \max \set {\norm {p^{-\paren{i + 1} } }_p, \norm {p^{-\paren {i + 1} } }_p}$

$\ds $

$=$

$\ds \norm {p^{-\paren {i + 1} } }_p$

$\ds $

$\le$

$\ds \norm {p^{-j } }_p$

$\ds $

$<$

$\ds \epsilon$

It follows that $\sequence {\alpha_n - \beta_n}$ is a null sequence by definition.

$\blacksquare$