Show that the Haar-Fourier series for \(u \in C_{c}\) converges uniformly for every \(x\) to \(u(x)\). Show that this remains true for functions \(u \in C_{\infty}\), i.e. the set of continuous functions such that \(\lim _{|x| ightarrow \infty} u(x)=0\).

[use the fact that \(u \in C_{c}\) is uniformly continuous. For \(u \in C_{\infty}\) observe that \(C_{\infty}=\bar{C}_{c}^{\|\cdot\|_{\infty}}\) (closure in sup-norm) and check that \(\left|s_{N}(u ; x)ight| \leqslant\|u\|_{\infty}\).]