# Thread: uniformly convergent of sequence of integrals

1. ## uniformly convergent of sequence of integrals

Let $f \in C(\mathbb{R})$ and $M>0$. Prove that the sequence of functions:
$f_n(z)=\frac{n}{2} \int^{z+\frac{1}{n}}_{z-\frac{1}{n}} f(y) dy$ is uniformly convergent to $f$ in $[-M,M]$.

2. ## Re: uniformly convergent of sequence of integrals

You can show, using the substitution $u=nt$ in $\frac n2\int_{-\frac 1n}^{\frac 1n}f(z+t)dt$ to get $f_n(z)=\frac 12\int_{-1}^1f\left(z+\frac un\right)du$. Now, use the fact that a continuous function on a compact set is uniformly continuous.