Let $\displaystyle f \in C(\mathbb{R}) $ and $\displaystyle M>0$. Prove that the sequence of functions:

$\displaystyle f_n(z)=\frac{n}{2} \int^{z+\frac{1}{n}}_{z-\frac{1}{n}} f(y) dy $ is uniformly convergent to $\displaystyle f$ in $\displaystyle [-M,M] $.