Pointwise convergence resisting averaging 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.
 A: That $C[0,1]$ fails the weak Banach Saks property appears as problem 17 in chapter VII in Joseph Diestel's Sequences and Series in Banach Spaces.
The problem gives an outline:
1) For $k$ a fixed positive integer, construct a nonnegative sequence $(g_n^k)_n$ in $B(C[0,1])$ satisfying:


*

*$g_n^k(t)=0$ if $t\notin ((k-1)/k, k/(k+1))$.

*$(g_n^k)$ converges pointwise to 0 on $[0,1]$.

*If $n_1<n_2<\cdots<n_k$, then there is an $a\in[0,1]$ with $g_{n_1}^k(a)=g_{n_2}^k(a)=\cdots=g_{n_k}^k(a)=1$.


2) Define $f_n=g_n^1+g_n^2+\cdots+g_n^n$. Show that:


*

*$(f_n)$ is weakly null in $C[0,1]$ (which is equivalent to saying $(f_n)$ converges pointwise to 0 and is bounded).

*If $n_1<n_2<\cdots<n_m<n_{m+1}<\cdots< n_{2m}$, then $(f_{n_1}+\cdots f_{n_{2m}})(t)\ge {1\over 2}$ for all $t\in[0,1]$ 
(I believe Diestel has a typo here, it should be $\ge m/2$ for some $t\in[0,1]$).



This was first proved by J. Schreier in Ein Gegenbeispiel zur Theorie der schwachen Konvergence, Studia Math. 2 (1930), 58–62.
 H. P. Rosenthal's article in volume 2 of  Handbook of the Geometry of Banach Spaces (proposition 4.21, page 1585) may also be helpful.
