 Supermartingale convergence with summable adapted drift

Suppose $X_n\ge 0$ is integrable and adapted, and $Y_n\ge 0$ is finite and adapted. Use the stopped predictable budget for adapted increments with an integer budget $a$. The process $X_{n\wedge {\tau }_a}+a-U_{n\wedge {\tau }_a}$ is an integrable nonnegative supermartingale, because its budget decrement cancels the localized drift. It has a finite almost-sure limit. On the event that the total drift is smaller than $a$, the stopping time is infinite, so $X_n$ itself converges. Taking the countable union over integer budgets proves finite almost sure convergence. No expectation of the total drift is needed.