Equivalence Class in P-adic Integers Contains Unique Coherent Sequence/Lemma 2

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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:

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


Proof

For all $j \in \N$:

\(\displaystyle \norm{\alpha_{j + 1} - \alpha_j}_p\) \(=\) \(\displaystyle \norm{\alpha_{j + 1} - \gamma_{j + 1} + \gamma_{j + 1} - \gamma_j + \gamma_j - \alpha_j}_p\)
\(\displaystyle \) \(\le\) \(\displaystyle \max \set { \norm{\alpha_{j + 1} - \gamma_{j + 1} }_p \mathop , \norm{\gamma_{j + 1} - \gamma_j }_p \mathop , \norm{\gamma_j - \alpha_j}_p}\)
\(\displaystyle \) \(\le\) \(\displaystyle \max \set { p^{-\paren{j + 2} } , \norm{\gamma_{j + 1} - \gamma_j }_p \mathop , p^{-\paren{j + 1} } }\) Assumption on $\sequence{\alpha_n}$
\(\displaystyle \) \(\le\) \(\displaystyle \max \set { p^{-\paren{j + 2} } , p^{-\paren{j + 1} } , p^{-\paren{j + 1} } }\) Assumption on $\sequence{\gamma_n}$
\(\displaystyle \) \(=\) \(\displaystyle p^{-\paren{j + 1} }\) Power Function on Integer between Zero and One is Strictly Decreasing

$\blacksquare$