Is there a bounded real-valued sequence with divergent Cesaro means (i.e. not Cesaro summable)?
More specifically, is there a bounded sequence $\{w_k\}\in l^\infty$ such that $$\lim_{M\rightarrow\infty} \frac{\sum_{k=1}^M w_k}{M}$$ does not exist?
I encountered this problem while studying the limit of average payoffs criterion for ranking payoff sequences in infinitely repeated games.