1. The problem statement, all variables and given/known data this is probably a dumb question, but i'm doing this proof where i have to show two sets are equal, where each set is a union from 1 to n sets. this is pretty easy to show with induction, i think, but i'm used to using induction when i have an infinite amount of things, so i'm not sure i'm allowed to use induction. any thoughts? specifically, it goes like this: Suppose that A_1, ..., An are Borel sets, that is they belong to ß. Define the following sets: B_1 = A_1, B_n = A_n ∩ (A_1∪ ... ∪ A_n-–1)^c (^c is complement), and let S equal the universal set. Show that U_i=1 to n A_i = U_i=1 to n B_i. 2. Relevant equations 3. The attempt at a solution U_1 to 1 A_i = A_1 = U_1 to 1 B_i = B_1. So we have a base case. So assume it's true for n=k. Then we have that U_i=1 to k A_i = U_i=1 to k B_i. Then we have that U_i to k B_i U B_k+1 = U_i to k A_i U (A_k+1 ∩ (A_1∪ ... ∪ A_k)^c =U_i to k A_i U (A_k+1 ∩ A_1^c ∩ A_2^c...∩A_k^c)... Let A_1^c ∩ A_2^c...∩A_k^c = D, and let U_i to k A_i = E Then we have U_i to k B_i U B_k+1 = E U (A_k+1 ∩ D) = (E U D) ∩ (E U A_k+1) = S ∩ (U_i to k A_i U A_k+1) = U_i=1 to k+1 A_i. god that looks hideous. hopefully it makes sense. any comments would be appreciated.