Let xn be adapted to fn with 0 le xnnbsple 1 let alpha beta

Question

Problem:

Let (X_n) be adapted to (F_n) with 0 ≤ X_n ≤ 1. Let α, β > 0 with α + β = 1. Suppose X₀ = x₀ and P(X_n+1 = α + βX_n|F_n) = X_n, P(X_n+1 = βX_n|F_n) = 1 - X_n

Show that X_n → X_∞ a.s., where P(X_∞ = 1) = x₀ and P(X_∞ = 0) = 1- x₀

Additional Information:

This problem is basically from Mathematics as well as it is about martingale calculation