Q) $\{X_n\}$,$\{Y_n\}$ are positive integrable and adapted to $\mathcal{F_n}$. Suppose $E(X_{n+1}|\mathcal{F_n})\leq X_n+Y_n$, with $\sum Y_n < \infty$ a.s. Prove that $X_n$ converges a.s. to a finite limit.
The first part of the answer is: Consider $W_{n}= X_n - \sum_{k=1}^{n}Y_k$ and the claim is that it is super martingale which I don't get.
$$E[W_{n+1}|\mathcal{F_n}] = E[X_{n+1}|\mathcal{F_n}]-\sum_{k=1}^{n+1}E[Y_k|\mathcal{F_n}] \leq X_n-\sum_{k=1}^{n}E[Y_k|\mathcal{F_n}] + Y_n-E[Y_{n+1}|\mathcal{F_n}] = $$ $$W_n+Y_n-E[Y_{n+1}|\mathcal{F_n}]$$
So I don't understand why $Y_n-E[Y_{n+1}|\mathcal{F_n}] \leq 0$? It is not given that $\{Y_n\}$ is a sub martingale. Thanks and appreciate a hint.
