I have learnt that the means of a convergent sequence also converges. However, I am wondering whether can we say the means of a bounded sequence converge? or any other counterexample?
Let $(a_n)_{n\in\mathbb N}$ be a bounded sequence. Then $\displaystyle \overline a_n=\sum_{k=1}^n \frac{a_k}n$ converges?
The following is my thinking so far. As $(a_n)_{n\in\mathbb N}$ be a bounded sequence, $\displaystyle \overline a_n$ is bounded as well, if it is monotonic then it converges. But $\displaystyle \overline a_n$ is not always monotonic.
I also tried Cauchy criterion, but it is massy to deal with averages.
