$(X_k)_k,(Y_k)_k,(W_k)_k$ are three $(\mathcal{F}_k)$-adapted processes taking values in $\mathbb{R}^+$ such that for all $k,$ $$E[X_{k+1}|\mathcal{F}_k] \leq X_k+Y_k-W_k \ \ \ \ \ \ \ \ (1)$$
We want to prove that on $\{\sum_{k}Y_k<\infty\},X_k$ converges a.s to a finite random variable $X$. To do so we consider: $U_k=X_k-\sum_{p=0}^{k-1}(Y_p-W_p),$ so that $U_k$ is a supermartingale.
In order to well define $E[U_{k+1}|\mathcal{F}_k],U_k$ should be positive or integrable.
What do suggest to do in this case ? Truncating each processes so that $(1)$ still holds?