Sequence of P-adic Integers has Convergent Subsequence/Lemma 5

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:
 * the canonical expansion $\ldots \, b_n \, \ldots \, b_1 b_0$ converges to some $x$ in the $p$-adic integers $\Z_p$

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

Then:
 * the subsequence $\sequence{x_{n_r}}_{r \in \N}$ converges to $x \in \Z_p$