Let \(u(x)=\mathbb{1}_{[0,1 / 3)}(x)\). Prove that the Haar-Fourier series diverges at \(x=\frac{1}{3}\).

[ verify that \(\lim \inf _{N ightarrow \infty} s_{N}\left(u, \frac{1}{3}ight)<\lim \sup _{N ightarrow \infty} s_{N}\left(u, \frac{1}{3}ight)\).]