SCHWEITZER 1985 Problem 12

Let $(\Omega, \mathcal A, P)$ be a probability space, and let $(X_n, \mathcal F_n)$ be an adapted sequence in $(\Omega, \mathcal A, P)$ (that is, for the $\sigma$ -algebras $\mathcal F_n$ , we have $\mathcal F_1\subseteq \mathcal F_2\subseteq \dots \subseteq \mathcal A$ , and for all $n$ , $X_n$ is an $\mathcal F_n$ -measurable and integrable random variable). Assume that $$\mathrm E (X_{n+1} \mid \mathcal F_n )=\frac12 X_n+\frac12 X_{n-1}\,\,\,\, (n=2, 3, \ldots )$$ Prove that $\mathrm{sup}_n \mathrm{E}|X_n|<\infty$ implies that $X_n$ converges with probability one as $n\to\infty$ . [I. Fazekas]