Martingale convergence

Question

I'm working on the same problem on this page except for the fact that my conditional expectation is $E[X_{n+1}|F_{n}]\leq X_{n}+Y_{n}$

I can't find a RV Z to recover a supermartingale from the equation..

I need some help..

Let $X_n$ and $Y_n$ be positive integrable and adapted to $F_n$. Suppose that $E[X_{n+1}|F_{n}]\leq X_{n}+Y_{n}$ and that $\sum_{n\geq 1}{Y_n} < \infty$. Prove that $X_n$ converges a.s. to a finite limit

Answer 1

Using the notation of your previous question, try $S_n=X_n-\sum\limits_{k\leqslant n-1}Y_k$ or $S_n=X_n+\sum\limits_{k\geqslant n}Y_k$.