Let's say I have a sequence $\{p_n\}$, whose range is the set $E$. Let's say $E$ contains a sequence $\{s_n\}$. Is $\{s_n\}$ necessarily a subsequence of $\{p_n\}$?
The definition of subsequences I must abide by is as follows: For any sequence $\{p_n\}$ in a metric space $X$, let $\{n_k\}$ be a sequence of natural numbers, with $n_1 < n_2 < \ldots < n_k$. Then the sequence given by $\{p_{n_k}\}$ is a subsequence of $\{p_n\}$.