P-adic Integer has Unique Coherent Sequence Representative/Lemma 1

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:

In either case:
 * $\norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren{j + 1} }$

Since $j$ was arbitrary, then:
 * $\forall j \in \N: \norm {\beta_{\map N {j + 1} } - \beta_{\map N j} }_p \le p^{-\paren {j + 1} }$