P-adic Integer has Unique Coherent Sequence Representative/Lemma 1
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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 $\sequence {\beta_n}$ be a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$ such that:
- $\forall j \in \N : \exists \mathop {\map N j} \ge j : \forall m, n \in \N: m, n \ge \map N j: \norm {\beta_n - \beta_m} \le p^{-\paren {j + 1} }$
Then:
- $\forall j \in \N: \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$
Proof
Let $j \in N$
Suppose $\map N {j + 1} \ge \map N j$
By definition:
- $\norm{\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$
Now suppose $\map N j \ge \map N {j + 1}$
Then:
\(\ds \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p\) | \(\le\) | \(\ds p^{-\paren {j + 2} }\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds p^{-\paren {j + 1} }\) | Power Function on Integer between Zero and One is Strictly Decreasing |
In either case:
- $\norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren{j + 1} }$
Since $j$ was arbitrary, then:
- $\forall j \in \N: \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$
$\blacksquare$