Subsequential Limits

Question

I'm working through Rudin's PoMA at the moment, and I've been learning about subsequential limits. However, I'm somewhat confused and I have a question, which is more conceptual than an actual exercise.

I know that when a sequence converges the $\lim \space \sup$ and $\lim \space \inf$ are equal to the $\lim$.

But when the sequence diverges to negative or positive infinity, shouldn't the only subsequential limit be negative or positive infinity, respectively?

So my question is: is the $\lim \space \sup/\inf$ concept only useful for sequences that oscillate around values(like $a_n=(-1)^n$) ? Is it ever useful for any other sequences?

Answer 1

Regarding your first question: yes, if the sequence has a limit (even infinity or minus infinity), then the $\limsup$ and $\liminf$ will agree with that.

The reason why $\limsup$ and $\liminf$ are useful is because, for real sequences, they always exist. So many things can be phrased using them, irrespective of whether you have a convergent sequence or not. Something that happens from time to time is that the proof that some limit exists consists in showing that $\limsup=\liminf$.

Answer 2

Well, it's also interesting to contemplate all possible limits of convergent subsequences. A good exercise is to prove that $\limsup a_n$ is the $\sup$ of all those possible limits.

For example, if you enumerate the rational numbers in a sequence $\{a_n\}$ one way or another, what will the set of all subsequential limits be? What will be the $\limsup$ and $\liminf$?