Some others concepts you will be discussing at the beginning of the course are: limit superiors (a.k.a – supremum limits, lim sup) and limit inferiors (a.k.a – infimum limits, lim inf). During the course you will be primarily discussing the lim sup and lim inf in the context of sets. You may have already seen examples of finding lim sup and lim inf in the context of sequences. The ideas are very similar and in the included video, both applications of lim sup/lim inf will be explored.
This section of the online supplement provides the following materials: a video about lim sup/ lim inf, and a couple of practice problems. In the video there are multiple in-video questions (be sure to carefully read the questions, some of them require you to type out a numerical answer, i.e. – you would need to type out “zero” not “0” – without the quotation marks). All of which involve finding either the lim sup/ lim inf of some sequence or set. I encourage you to challenge yourself, and find both the lim sup and the lim inf (or the sup and inf). In this section you will notice that the content and problems are not very proof-oriented, in that you are not being asked to prove anything. You may encounter problems like this throughout the course as well, but notice that even in this section of material more proof-oriented problems can still be asked.
There are practice problems that you can try out for yourself. One is related to sequences, and the other is related to sets. If you would like to just see the practice problems (without the solutions), so you can try working through it yourself, click here. The solutions to the practice problems is also provided here.