(Pah! Some noders just don't pick up their nodeshells after the party. I guess somebody has to do it...)
We want to exhibit a bounded sequence with no Cesaro limit. Consider the sequence 100111100000000..., i.e. a one, then 2 zeros, then 4 ones, then 8 zeros etc.
Now for odd n, the sum of the first 2n elements contains at least 2n-1 + 2n-3 ones, namely from the last block and the block two steps earlier. So the average value of the first 2n elements is ≥ (2n-1 + 2n-3)/2n = 5/8. Similarly, for even n the average is ≤ 3/8. This shows that the average value of the sequence does not converge to a limit.