Let $f$ satisfies $$|f(x+u) - f(x)|\leq L|u|^{\alpha}$$
for some constants $L$ and $\alpha$. If $\alpha = 1$ then $f$ is called Lipschitz continuous, and if $0 < \alpha < 1$ then $f$ is Hölder continuous. Show that if $f$ is $2\pi$-periodic and Lipschitz or Hölder continuous, then its Fourier series converges to $f(x)$ for each $x$.