$\lim_{i\to\infty} a_i = 0 \implies \lim_{n\to\infty} \sum_{i=1}^n \frac{a_i}{n} = 0$?

Question

Let $a_i$ be a real number. Suppose that $\lim_{i\to\infty} a_i = 0$. Is it true that $\lim_{n\to\infty} \sum_{i=1}^n \frac{a_i}{n} = 0$?

Answer 1

Yes, this is a consequence of the cesaro lemma.

http://www.ee.columbia.edu/~vittorio/CesaroMeans.pdf