Found problems: 1
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]