Sequence of P-adic Integers has Convergent Subsequence/Lemma 4

Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Let $\sequence{x_n}$ be a sequence of $p$-adic integers.

Let $\sequence{b_n}$ be a sequence of $p$-adic digits such that:
 * for all $j \in \N$, there exists infinitely many $n \in \N$ such that the canonical expansion of $x_n$ begins with the $p$-adic digits $b_j \, \ldots \, b_1 b_0$

Then:
 * there exists a subsequence $\sequence{x_{n_rj}}_{j \in \N}$ of $\sequence{x_n}$:
 * for all $j \in \N$, the canonical expansion of $x_{n_j}$ begins with the $p$-adic digits $b_j \, \ldots \, b_1 b_0$