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 $j \in \N$: