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

Proof
For all $j \in \N$: