(If needed, reference book: http://math.sfsu.edu/beck/papers/aop.noprint.pdf (Chapter 5/6))

1. (Limits of sets). Let X be a universal set, and let Ai C X be sets indexed by i E N. We call such a family of sets a sequence of sets. Define lim sup A; := n UA; jeN i>j lim inf A := UNA. jeNiZj We say that the sequence of sets {A : N} has a limit if lim sup Ai = lim inf A. In this case, we say the limit exists, and write simply lim A = lim sup Ai = lim inf Ai. (a) Show that lim inf A; C lim sup A. (b) We call a sequence of sets {Aj} non-increasing if Ai+1 C A for all i E N. Similarly, we call such a sequence of sets {Aj} non-decreasing if Ai C Ai+1. Show that if {Aj} is either non-increasing or non-decreasing that lim inf Ai = lim sup Ai, i.e. that lim Ai exists. (c) Give an example of a sequence of sets {Aj} with A; C Z which has a limit. (d) Give an example of a sequence of sets {Aj} with Ai C Z which does not have a limit. 1. (Limits of sets). Let X be a universal set, and let Ai C X be sets indexed by i E N. We call such a family of sets a sequence of sets. Define lim sup A; := n UA; jeN i>j lim inf A := UNA. jeNiZj We say that the sequence of sets {A : N} has a limit if lim sup Ai = lim inf A. In this case, we say the limit exists, and write simply lim A = lim sup Ai = lim inf Ai. (a) Show that lim inf A; C lim sup A. (b) We call a sequence of sets {Aj} non-increasing if Ai+1 C A for all i E N. Similarly, we call such a sequence of sets {Aj} non-decreasing if Ai C Ai+1. Show that if {Aj} is either non-increasing or non-decreasing that lim inf Ai = lim sup Ai, i.e. that lim Ai exists. (c) Give an example of a sequence of sets {Aj} with A; C Z which has a limit. (d) Give an example of a sequence of sets {Aj} with Ai C Z which does not have a limit