Question: Let $a_n$ be a sequence of real numbers. Is it true that for every $\varepsilon > 0$, if
$$\left \lvert \frac{1}{N} \left ( \sum_{n=0}^{N-1} a_n \right )\right \rvert < \frac{1}{N^{1+\varepsilon}}$$
for all $N \in \mathbb Z_+$, then $a_n$ converges to $0$?
Best Answer
Multiply by $N$ to get that
$$\left \lvert \sum_{n=0}^{N-1} a_n \right \rvert < \frac{1}{N^{\varepsilon}}$$
Therefore $\sum_{n=0}^\infty a_n=0$ and $\lim_{n\to\infty} a_n=0$.