Can you give an example of a sequence of continuous functions $f_n:[0,1]\to [0,1]$, such that $f_n\to 0$ pointwise and there is no subsequence $(f_{n_k})$ for which $\frac 1 m\sum_{k=1}^{m}f_{n_k}$ tends to zero uniformly?
I think it's the same as asking whether the Banach space of continuous real valued functions has the "weak Banach-Saks property", but I was unable to find out the answer.