TITLE: Bounded sequence with divergent Cesaro means
QUESTION [10 upvotes]: 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.

REPLY [2 votes]: Let $(a_n)$ be any increasing sequence of positive integers such that
$$
a_{n+1}\ge 2(a_1+\cdots+a_{n})
$$
for all $n$. Now let $(x_n)$ be the sequence 
$$
\underbrace{0,\ldots,0}_{a_1 \text{ times }},\underbrace{1,\ldots,1}_{a_2 \text{ times }},\underbrace{0,\ldots,0}_{a_3 \text{ times }},\underbrace{1,\ldots,1}_{a_4 \text{ times }},\ldots
$$
It easily follows that
$$
\frac{x_1+\cdots+x_{a_1+\cdots+a_{2n}}}{a_1+\cdots+a_{2n}}\ge \frac{a_{2n}}{a_1+\cdots+a_{2n}}=1-\frac{a_1+\cdots+a_{2n-1}}{a_1+\cdots+a_{2n}}\ge 1-\frac{1}{1+2}=\frac{2}{3}
$$
for all $n$ and, on the other hand,
$$
\frac{x_1+\cdots+x_{a_1+\cdots+a_{2n-1}}}{a_1+\cdots+a_{2n-1}}\le \frac{a_1+\cdots+a_{2n-1}}{a_1+\cdots+a_{2n}} \le \frac{1}{1+2}=\frac{1}{3},
$$
hence it cannot converge.