# Cesàro summability

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?

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 $$n\in I_j$$.)