SCHWEITZER 1975 Problem 8

Prove that if $$\sum_{n=1}^m a_n \leq Na_m \;(m=1,2,...)$$ holds for a sequence $\{a_n \}$ of nonnegative real numbers with some positive integer $N$ , then $\alpha_{i+p} \geq p \alpha_i$ for $i,p=1,2,...,$ where $$\alpha_i= \sum_{n=(i-1)N+1}^{iN} a_n \;(i=1,2,...)\ .$$

L. Leindler