Originally Posted by
emakarov If A is a set, let in(x, A) = 1 if x is in A and in(x, A) = 0 otherwise. Then lim (A_n) exists iff for every x, the limit of in(x, A_n) exists, i.e., in(x, A_n) eventually stabilizes. Also, x is in lim(A_n) iff in(x, A_n) stabilizes at 1. It is easy to express in(x, A * B) through in(x, A) and in(x, B) where * is each of the four set operations in the question. Using these facts, it is possible to show that the limits in the left-hand sides exist and the equalities hold.