Prove with steps, and the Kolmogorov 0-1 law is also in attached picture

Theorem 2.1. Let A1, A2, ... be independent events in (0, F, P). Then for any event Aco({ An, Anti,...}). 7=1 P(A) is either 0 or 1.Problem 4 (20p). Let (0, F, P) be a probability space and let A1, A2, ... be independent events in F. Let X : 0 -> R be a random variable (with o-algebras F and B), which is also a random variable when ? is equipped with the o-algebra o({An, An+1, ...}), for every n E N. Prove that there is a unique r ( R such that P((X = x}) = 1. You may assume the weaker form of the Kolmogorov 0-1 Law from Problem Set 3. Hint: Be careful, there are plenty of real valued random variables such that P((X = x}) =0 for all TER