I am trying to understand the notation and implication of this part of a proof from Theorem 3.6 of Principles of Mathematical Analysis by W. Rudin:
Theorem: Every bounded sequence in $R^k$ contains a convergent subsequence
Proof (partial): Let $E$ be the range of $\{p_n\}$. If $E$ is finite then there is a $p \in E$ and a sequence $\{n_i\}$ with $n_1 < n_2 < n_3 < ...$, such that $p_{n_1} = p_{n_2} = ... = p$. The subsequence $\{p_{n_i}\}$ so obtained converges evidently to $p$.
I'm having trouble understanding what this part of the proof is trying to say (but I'm pretty sure it's very simple).
If $E$ is finite, then the sequence $\{p_n\}$ is placing infinitely many things into finitely many bins. Suppose the range $E$ of the sequence $\{p_n\}$ is $E=\{p,q,r\}$. Since $\{p_n\}$ is infinite, at least one of $p,q,$ and $r$ (say $p$) must be hit infinitely many times by the sequence. Then we can take a subsequence $p_{n_1},p_{n_2},\ldots$ which is just $p,p,\ldots$. Certainly this subsequence converges to $p$.