Please solve the following question, and clearly write down the answers with steps. Don't copy others' answer. Thank you!

Problem 3 (20p). Let {Xi, Y, | i c N} be an independent collection of random variables on a probability space (0, F, P), each of which has the uniform distribution on [0, 1]. Let f : [0, 1] -> [0, 1] be continuous, and for each i c N set A, := {f(X.) > Y}EF and Z; :=1A,. Show that Zi a.8. f(x)dr as n -> 00