= Deterministic summable drift correction
{title2=$Z_n=X_n+\sum_{j\geq n}\delta_j$}
When the drift bound is a deterministic summable sequence $\delta_n\geq0$, the displayed correction is a nonnegative <supermartingale>. Its added tail tends to zero, so the <almost sure supermartingale convergence theorem> directly implies finite <almost sure convergence> of $X_n$. This simpler tail correction requires deterministic bounds; an arbitrary future random drift tail is not generally <measurable> at the current time.
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