(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Can you use induction on n cases (as opposed to infinity)?

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