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Element of Infinitely Many Subsets of a Given Set

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Let A_n be a sequence of subsets of a given set X, and let J be the set of all x in X that are in infinitely many A_n.

Show that J is equal to the intersection (from n = 1 to infinity) of [the union (from i = n to infinity) of A_i].

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Solution Summary

A complete, detailed proof of the claimed set equality is presented.