Fix $\alpha , \beta > 0$ with $\alpha + \beta = 1$ and consider the sequence of random variables defined by $X_0 = \theta \in (0,1)$ and
$P(X_n = \alpha + \beta X_{n-1} \mid X_{n-1} , ... , X_0) = X_n , \qquad P(X_n = \beta X_{n-1} \mid X_{n-1} , ... , X_0) =1- X_n.$
I am trying to prove that $P(\lim_{n\rightarrow \infty} X_n = 1) = \theta$ and $P(\lim_{n\rightarrow \infty} X_n = 0) = 1-\theta$. It seems that there should be a way to do this using the result that $P(E \mid \mathcal F_n) \rightarrow P(E \mid \mathcal F)$, whenever $\mathcal F_n \uparrow \mathcal F$ is an increasing sequence of $\sigma$ algebras. But, I am not seeing a solution. Any suggestions?
By the way, this is an exercise on page 224 of Durrett's Probability: Theory and Examples.