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Limits of sequences of sets Definition 1 Let (An)n∈N be a sequence of subsets of a set X. (a) We say that (An) is increasing if An ⊂ An+1 for all n ∈ N, and decreasing if An ⊃ An+1 for all n ∈ N. (b) For an increasing sequence (An), we define limn→∞ An := ∞ n=1 An. For a decreasing sequence (An), we define limn→∞ An := \∞ n=1 An. Definition 2 For any sequence (An) of subsets of a set X, we define lim inf n→∞ An := n∈N \ k≥n Ak lim sup n→∞ An := \ n∈N k≥n Ak. Proposition 1 Let (An) be a sequence of subsets of a set X. Then (i) lim inf n→∞ An = {x ∈ X : x ∈ An for all but finitely many n ∈ N}. (ii) lim sup n→∞ An = {x ∈ X : x ∈ An for infinitely many n ∈ N}. (iii) lim inf n→∞ An ⊂ lim sup n→∞ An