Cesàro summability

Question

Suppose $(a_n)$ is a Cesàro summable sequence of positive real numbers (i.e., $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n a_i$ exists and is finite) and $(b_n)$ is a bounded sequence of positive real numbers. Is the sequence $(c_n)$ defined by $c_n=a_nb_n$ also Cesàro summable?

Answer 1

This is clearly false. For example, let $a_n=1$ and let $(b_n)$ be any bounded sequence that is not Cesaro summable.

(To be specific, say $I_j=[2^j,2^{j+1})$ and define $b_n=(-1)^j$ for $n\in I_j$.)