Cesaro means $\frac{1}{N} \sum_1^{N} a_n$ of a positive sequence converging to zero while the sequence itself does not

Question

In this post: Cesaro means of a positive sequence, OP is referring to a "famous example" of a sequence $a_n \geq 0$ whose Cesaro means $\frac{1}{N} \sum_1^{N} a_n$ converge to zero while the sequence itself does not. What "famous example" is OP referring to?

Answer 1

Take $$a_k = 1 \quad \text{ if } k=2^n \text{ for a }n \in \mathbb{N} \quad \quad \text{and} \quad a_k=0 \text{ otherwise}$$