Noneffectiveness of a class of regular matrices

We show that if a sequence $\{\varepsilon_n\}$ is such that $\varepsilon_1>\varepsilon_2\ge\varepsilon_3\ge\dots$, $\sum_{n=1}^\infty\varepsilon_n=1$, then for any bounded sequence $\{S_n\}$ the equation $\lim\limits_{n\to\infty}\sum_{k=1}^n\varepsilon_{n+1-k}S_k=S$ implies the equation $\lim\limits_{n\to\infty}S_n=S$. This theorem generalizes a theorem of N. A. Davydov [2].