In the rest of the second part, $f$ is an element of $\mathcal{C}_{0}$. For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by
$$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$
where, for all $(j, k) \in \mathcal{I}$,
$$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$
Show that $\lim_{j \rightarrow +\infty} \max_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| = 0$.