Renyi's proof of Faddeev's theorem: convergence of a series

Question

How can we prove that$$ \lim_{m\rightarrow \infty} \sum_{k=0}^{m} \frac{\delta_{k}}{m}=0$$ given $\lim\limits_{k\rightarrow \infty}\delta_{k}=0$?

Answer 1

Let $\varepsilon$ be given. $\exists M (m>M\implies |\delta_n|<\varepsilon)$. Choose $m$ so large that $$m>M \text{ and } \frac{1}{m} \left\lvert\sum_{k=0}^M{\delta_k}\right\rvert<\varepsilon$$ Then $$\frac{1}{m}\left\lvert\sum_{k=0}^m{\delta_k}\right\rvert\le \frac{1}{m} \left\lvert\sum_{k=0}^M{\delta_k}\right\rvert+\frac{1}{m} \left\lvert\sum_{k=M+1}^m{\delta_k}\right\rvert\le\frac{M\varepsilon}{m}+\varepsilon<2\varepsilon $$