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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?

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Regarding your first question: yes, if the sequence has a limit (even infinity of 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 $\limsup=\liminf$.

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  • $\begingroup$ Thanks for the quick response! I understand the second part of your answer but unfortunately not the first. The sequence you created from integers and sequences A and B doesn't diverge to infinity, right? For some M there cannot be an N such that n>N implies the sequence is greater than M, because the terms from A and B will eventually be near 0 and 1. It seems that the sequence is an oscillating type sequence, bouncing between large negative, large positive, and terms of A and B. Sorry if I'm using the wrong terminology. $\endgroup$ – Robearz Jun 1 '13 at 19:32
  • $\begingroup$ If a sequence $\{c_n\}$ diverges to infinity, then $\lim c_n=\limsup c_n=\liminf c_n=\infty$. Sorry, I misread your question above. $\endgroup$ – Martin Argerami Jun 1 '13 at 19:34
  • $\begingroup$ I have edited the answer accordingly. $\endgroup$ – Martin Argerami Jun 1 '13 at 19:36
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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 $\liminf$?

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