I have a sequence of continuous (even analytic) functions on $[0,1]$. I know that the Cesaro mean of the sequence of functions converges uniformly on $[0,1]$: $$\frac{1}{N}\sum_{n=1}^Nf_n\rightarrow f$$ I was also able to show that this sequence is uniformly Lipschitz and bounded, so I know that there exists a subsequence of functions which converges uniformly on $[0,1]$ (using Arzela-Ascoli). I'd like to determine if there possibly exists a subsequence of functions which converges uniformly to the Cesaro mean itself ($f$). So basically I'd be happy to hear some ideas about possible criterions I could use. I am obviously not saying that there should always be a subsequence which converges to the Cesaro mean, but maybe there are some useful special cases which I can use.
At first I tried to possibly think in the direction of the Banach-Saks theorem and weak-$*$ topology. I know that the unit ball in $C([0,1])$ is weak-$*$ compact. I am actually not sure if it is weak-$*$ sequentially compact, but let's say it is. Then my bounded sequence contains a subsequence which converges in the weak-$*$ topology. Then I could maybe apply some version of Banach-Saks to show that this subsequence has a smaller subsequence which actually converges to the mean value in norm (so uniformly). But as I said, I am not sure if there is a version of Banach-Saks which applies in this case, and I am overall not sure about the details and if this should work, this was just a hunch.
Anyway, I'd be happy to hear some ideas as to what tools might come in handy and under what assumptions I can actually show something like this.
Thanks in advance.