Minimum-decrement convergence criterion ($y_{n+1}\le y_n-\lambda \min (x_n,y_n)$)

Suppose $x_n,y_n>0$ and the displayed inequality holds with fixed $\lambda >0$. Then $y_n$ decreases to some $l\ge 0$. Telescoping shows ${\sum }_n\min (x_n,y_n)\le y_1/\lambda$. If $l>0$, this bounds ${\sum }_n\min (x_n,l)$, contradicting clipping preserves divergence of a positive series when ${\sum }_n{x}_n$ diverges. Thus divergence of the positive input series forces $y_n\to 0$.