Minimum-decrement convergence criterion
= Minimum-decrement convergence criterion
{title2=$y_{n+1}\le y_n-\lambda\min(x_n,y_n)$}
Suppose $x_n,y_n>0$ and the displayed inequality holds with fixed $\lambda>0$. Then $y_n$ decreases to some $l\ge0$. 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 $\sum_nx_n$ diverges. Thus divergence of the positive input series forces $y_n\to0$.