For a continuous function $f$ of period $1,$ how to prove that $$\lambda_{n}=\frac{1}{n}\sum_{k=0}^{n-1}S_k(f),~~~~n \geq 1,$$ converges to $f$ pointwise, where $S_{k},~k \geq 0$ are partial sums of the Fourier series of $f.$
Starting with the Fourier series itslef, $f=\sum_{n \in \mathbb{Z}} c_n e^{i n t},$ where $c_n=\frac{1}{2 \pi} \int_{-\infty}^{\infty} f(t) \cdot e^{-int}~dt,$ and the partial sum is deifined accordingly. Does this have to do with a double sum and where do I need to use the periodicity, $f(t+1)=f(t)$ for all $t$ ? Is there a way to do this problem without using the Dirichlet kernel ? Any help in solving this problem is appreciated.