Find a convergent subsequence in the sequence of rational primes in $\mathbb{Q}_p$

Question

In this post, my main question is how to solve the following problem:

Consider the sequence of primes $2, 3, 5, 7, 11, \ldots$ in the $p$-adic field $\mathbb{Q}_p$, find a convergent subsequence.

My attempts: Since $\mathbb{Q}_p$ is complete, it is enough for us to construct a subsequence which is a Cauchy sequence. So after all, I have to obtain an estimate of the gaps between primes (in the $p$-adic sense). Namely, arbitrarily picking two primes $p_{m}$ and $p_{n}$, we need to estimate the gap $$\mathrm{Gap}(m,n) := \vert p_m - p_n \vert_p.$$ By the non-archimedian-ness of the absolute value $\vert\cdot \vert_p$, we have $$ \mathrm{Gap}(m,n) := \vert p_m - p_n \vert_p \leq \max\{ \vert p_m \vert_p, \vert p_n \vert_p \} = 1 $$ if one of the primes $p_m$ is not $p$ itself. But this wild estimate is far from enough. I'm been stucking here and making no progress besides this.

Thank you all for commenting and answering this question! :)

Answer 1

HINT: Let $P_0$ be the set of primes. Given an infinite $P_k\subseteq P_0$ for some $k\ge 0$, there is an $r_{k+1}\in\{1,\ldots,p^{k+1}-1\}$ such that

$$P_{k+1}=\{n\in P_k:p_n\equiv r_{k+1}\pmod{p^{k+1}}\}$$

is infinite. Now use the sets $P_k$ to construct your Cauchy sequence.

Answer 2

Define $a_1 :=$ the smallest prime $\neq p$, and if we have $a_m$, look at the set $\lbrace a_m + k p^{m+1} : k \in \mathbb N\rbrace$. By Dirichlet, it contains a prime (actually infinitely many); choose one of them which is $\neq p$, call it $a_{m+1}$.

The sequence $(a_i)_i$ is a subsequence of the sequence of all primes, and by definition, $\lvert a_m -a_{m+1}\rvert_p \le p^{-m-1}$.