Prove the uniform convergence of Fourier series

Question

Suppose $f \in C^1$ with a period of $2\pi$. Its Fourier series: $$ f \sim \sum_{n=-\infty}^{\infty} \widehat{f}(n) \mathrm{e}^{\mathrm{i} n t} $$ I want to prove that: $$ \left\|f-S_{N}(f)\right\|_{\infty} \leqslant \sqrt{\frac{2}{N}}\left\|f^{\prime}\right\|_{2}, \quad \forall N \geqslant 1 $$ where $S_{n}(f)(t)=\sum_{k=-n}^{n} \widehat{f}(k) \mathrm{e}^{\mathrm{i} k t}$, which means the uniform convergence of Fourier series.

Answer 1

Note that $$f(t)-S_N(f;t)\sim\sum_{|n|>N}\hat f(n)e^{int},$$while $$\sum_{|n|>N}|\hat f(n)|\le\left(\sum_{|n|>N}n^{-2}\right)^{1/2} \left(\sum_{|n|>N}n^2|\hat f(n)|^2\right)^{1/2}=\sqrt{\frac2N}||f'||_2.$$