An example in set theory of sets given as an infinite union or intersection

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This is an example of sets given as an infinite union or intersection of sets, and how to use set theory to compute a simplified description of such sets. #math #settheory #union #intersection
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I'm a bit confused about the end of part (a) of the proof. If n is an integer such that m = 2n, I'd say that m is in A_|n|, because n can be negative. Why are we sure that m is in A_m?

paoloscarpat