Well, I am learning about Fourier sum, and I encountered Cesaro sum in the proof of convergent uniformly of Fourier sum, I know that Fejér sentence says that: $\|f(x) - \sigma_n(f))\| < \epsilon .$ for all x.
then I know I should use Cesaro sentence that says: $\lim_{n\to\infty}{s_n(f)}=L,$ than Cesaro sum also $\lim_{n\to\infty}{\sigma_n(f)}=L$
but then I see that I should prove the convergent between Cesaro sum and the partial sum
$\|s_n(f) - \sigma_n(f))\| < \dfrac 1{N+1}\sum_{k=-N}^N|n||\hat{f}(n)| < \epsilon$
which add the condition with Riemann-Lebesgue Lemma that $f$ should be continuously differentiable to have Fourier series converge uniformly.
which confuse me because I thought Cesaro sentence already promise me that $\|s_n(f) - \sigma_n(f))\| < \epsilon$ , so why should I check when the sums converge if I have the sentence that's promise me that, or I misunderstood the sentence.