P-adic Integer has Unique Coherent Sequence Representative/Lemma 2

Theorem

Let $p$ be a prime number.

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

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

$\forall j \in \N: \norm {\gamma_{j + 1} - \gamma_j }_p \le p^{-\paren {j + 1} }$

Let $\sequence {\alpha_n}$ be a sequence in $\Q$ such that:

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

Then:

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

For all $j \in \N$:

$\ds \norm {\alpha_{j + 1} - \alpha_j}_p$

$=$

$\ds \norm {\alpha_{j + 1} - \gamma_{j + 1} + \gamma_{j + 1} - \gamma_j + \gamma_j - \alpha_j}_p$

$\ds $

$\le$

$\ds \max \set {\norm {\alpha_{j + 1} - \gamma_{j + 1} }_p \mathop , \norm {\gamma_{j + 1} - \gamma_j }_p \mathop , \norm{\gamma_j - \alpha_j}_p}$

$\ds $

$\le$

$\ds \max \set {p^{-\paren{j + 2} } , \norm{\gamma_{j + 1} - \gamma_j }_p \mathop , p^{-\paren {j + 1} } }$

Assumption on $\sequence{\alpha_n}$

$\ds $

$\le$

$\ds \max \set {p^{-\paren {j + 2} } , p^{-\paren {j + 1} } , p^{-\paren {j + 1} } }$

Assumption on $\sequence{\gamma_n}$

$\ds $

$=$

$\ds p^{-\paren {j + 1} }$

$\blacksquare$