I just can't understand the concept of $\lim \sup/\inf$ of sets, even after reading a lot about it. In limit infinum why do we have union sign with intersections (infima)? Can someone explain it very easily? Thank you very much.

  • $\begingroup$ No, sorry. I know there are a lot of questions like this, but i dont understand the answers $\endgroup$ Mar 1, 2020 at 22:19
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    $\begingroup$ It would help us to help you if you can say something about what you don't understand in these answers. Otherwise you're just going to get a whole lot of the same answers from those "a lot of questions like this". In fact, writing out exactly what you don't understand might just end up making you understand. $\endgroup$
    – Asaf Karagila
    Mar 1, 2020 at 22:34
  • $\begingroup$ I dont understand the definition. In lim inferior definition(union of intersactions) in wikipedia is said" So the limit infimum contains all subsets which are lower bounds for all except finitely many sets of the sequence." This is what i dont get.. $\endgroup$ Mar 1, 2020 at 22:44
  • $\begingroup$ But we are not asking what you don't understand on Wikipedia. We are asking what is unclear in the many answers that you say that you've read on this site. The more details you edit into your question, explaining exactly what confuses you, the easier it is to give you an answer (or point you at the direction of an older post). $\endgroup$
    – Asaf Karagila
    Mar 1, 2020 at 22:53
  • $\begingroup$ I already said. $\endgroup$ Mar 1, 2020 at 22:56

1 Answer 1


Suppose you have a sequence that represents a function of time. Suppose further that this sequence is the sum of two separate contributions: transient and steady state. The transient part of the signal is the part that has a limited time presence. The steady state part of the signal is the part that goes on forever.

For example, consider the following function $ f(x) = exp( -x / 10 ) + sin( x )$

Note that the exponential part of the signal dies away to $0$, while the sin part of the signal stays around forever.

Intuitively, limit superior and limit inferior are asking what are the supremum and infimum after all of the transient parts have died away.

  • $\begingroup$ Hm,thank you. But how i think about limit superior/inferior when i have intersection of union of sets? When looking in wikipedia : "So the limit supremum is contained in all subsets which are upper bounds for all except finitely many sets of the sequence". What is meant by that? $\endgroup$ Mar 1, 2020 at 22:39
  • $\begingroup$ Where in wikipedia does it say that? $\endgroup$ Mar 1, 2020 at 22:48
  • $\begingroup$ in Special case: discrete metric $\endgroup$ Mar 1, 2020 at 22:53
  • $\begingroup$ do you understand what the discrete metric is? There is a lot written in that section, so it would help if you could be more specific about which part of it you don’t undertand $\endgroup$ Mar 3, 2020 at 22:26

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