Let $f$ be such that $\int_{-\pi}^{\pi}f(x)dx=\int_c^{c+2\pi}f(x)dx \quad\forall c\in\mathbb{R}$ and suppose that $c\in[-\pi,\pi].\quad$ If $\Gamma_n(f)$ denotes the $n$th Fejér mean. Show that $$\text{lim} \Gamma_n(f)=\frac{1}{2}[f(c-)+f(c+)]$$
I know that the Fejér mean is defined by: $\Gamma_n(f) = \frac{1}{n}\sum_{k=0}^{n-1}S_n(f)$
$S_n(f)$ denotes the Fourier series of the function $f$
I've been struggling with this proof. Any suggestions would be great!