Let $\phi:{\bf R}_{+}\to{\bf R}_{+}$ be a decreasing right-continuous function with $\lim_{s\to\infty}\phi(s)=0$. Assume that $r\phi(s+r)\leq B_{0}\phi(s)^{1+\delta_{0}}$ for some constant $B_{0}>0$ and all $s>0$ and $r\in[0,1]$. Then there exists some $S_{\infty}=S_{\infty}(\delta_{0},B_{0},\phi)>0$ such that $\phi(s)=0$ for all $s\geq S_{\infty}$.