a bounded sequence with no Cesaro limit

(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 2^n-1 + 2^n-3 ones, namely from the last block and the block two steps earlier. So the average value of the first 2ⁿ elements is ≥ (2^n-1 + 2^n-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.