I'm working on the the construction of BM given by Lévy-Ciesielski. The author begin to prove another result and for this reason he assume that BM exists and that it is also differentiable. For this reason he comes to the point where he write that the derived process on $[0,1]$ is normal since limit of normal random variables. But I don't see why the derivative have to be centered and with $Cov(W_t',W_s')=0 \forall t\neq s$. Moreover he states that $\mathbb{E}((W_t')^2)=\mathbb{E}\left[\lim_{h\rightarrow 0}\left(\frac{W_{t+h}-W_{t}}{h}\right)^2\right]=\lim_{h\rightarrow0}\frac{h}{h^2}=\infty$.
So I think that my questions are all related with the problem of interchanging the limit. I don't see why the derivatives have to be centred, have covariance 0 and the reason we can interchange the limit in the above equality.