4
$\begingroup$

I am trying to understand whether the limit infimum of a set is related or a generalization of the limit infimum of a sequence of real numbers. Suppose that $X_n$ is a sequence of sets, and so the limit infimum of $X_n$ is defined as saying that from some stage onwards, all of the $X_n$ occur.

Concretely:

$$ \liminf_{n \to \infty}X_n = \bigcup_{n=1}^{\infty}\bigcap_{k\geq n}X_k $$

Now, suppose that we let $X_n(\omega) = (-1)^n$ and let $X(\omega) = 1$. Then, we have that:

$$ \liminf_{n \to \infty}X_n = -1 $$

Here it appears that after some time, only $-1$ occurs. But, I know that both $1$ and $-1$ alternate, and so there shouldn't be a time after which $-1$ only occurs.

I am wondering if I am somehow confusing something here between the limit of sets and the limit of a sequence? It also seems that the definition of the limit infimum/supremum of sets has no direct connection to that of a sequence and is an arbitrary definition. Am I wrong here?

$\endgroup$

marked as duplicate by Did limits Oct 29 '16 at 12:16

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Sorry, what's $\omega$? $\endgroup$ – yo' Oct 29 '16 at 11:58
  • $\begingroup$ @Brian M. Scott may be helpful here $\endgroup$ – ILoveMath Oct 29 '16 at 11:58
  • $\begingroup$ @yo' The other post mentions limsups. I am sure you will be able to explain how to transpose these explanations to liminfs. $\endgroup$ – Did Oct 29 '16 at 12:38
  • $\begingroup$ @yo' While we are exchanging views, please explain how you came to post an answer defining the liminf as a limsup. Did you check any reference at all on the subject before posting? $\endgroup$ – Did Oct 29 '16 at 12:39
  • $\begingroup$ after re-reading your comments and the other thread, I see my points are useless. I just wonder whether the OP will be satisfied with the other thread... Howgh. $\endgroup$ – yo' Oct 29 '16 at 12:43