= Supermartingale convergence with summable adapted drift
{title2=$\mathbb E[X_{n+1}\mid\mathcal F_n]\leq X_n+Y_n,\quad\sum_nY_n<\infty\text{ a.s.}$}
Suppose $X_n\geq0$ is <integrable> and adapted, and $Y_n\geq0$ 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.
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