Let $x$ be a sequence. $S(x)$ is the set of all finite subsequential limits of $x$. $S^{'}(x)$ is the set of all subsequential limits of $x$, that is $+\infty$ and $-\infty$ can also be included.
(i) Investigate whether $S(x)$ and $S^{'}(x)$ may be empty or not.
(ii) What are the possible cardinal numbers of $S(x)$ and $S^{'}(x)$?
(iii) Show that if $S(x)$ is not upper bounded then $+\infty \in S^{'}(x)$. Similarly, show that if $S(x)$ is not lower bounded then $-\infty \in S^{'}(x)$.
My attempt:
(i) $S(x)$ can be empty for example for $x_n=n$. But $S^{'}(x)$ is never empty. If $x$ is not bounded below, then $-\infty \in S^{'}(x)$. If $x$ is not bounded above, then $+\infty \in S^{'}(x)$. And if $x$ is bounded above and below, it must have a convergent subsequence, thus a subsequential limit.
(ii) The size of these two sets differ by at most 2, so analysing one of them is enough. $S(x)$ can have size $n$ for any finite $n$ since the sequence $1,2,...,n-1,n,1,2,...$ has subsequential limits $1,2,...,n$. It can also be countably infinite by looking at the sequence $1,1,2,1,2,3,1,2,3,4$. But I am unsure of uncountably infinite case, so I need help for this.
(iii) For this question, I am not sure how to begin. So a hint or a solution would be perfect.
