# Uniform Convergence and Integrals

• March 8th 2011, 03:54 PM
zebra2147
Uniform Convergence and Integrals
I'm having trouble getting started with this proof. Any help is appreciated...

If $f[0,1]\rightarrow \mathbb{R}$ is continuous, then, $\int _{0}^{1}f=lim_{n\rightarrow \infty}\sum_{k=1}^{n}f(\frac{k}{n})(\frac{1}{n})$.

My thinking is that I need to construct step functions for $f$ but I guess I don't know where to go with that.
• March 8th 2011, 05:45 PM
roninpro
Isn't the term on the right a Riemann sum using rectangles of uniform width $1/n$?
• March 8th 2011, 06:08 PM
zebra2147
So then the height of the rectangle would be defined by $f(\frac{x}{n})$ since $f$ maps $[0,1]\rightarrow \mathbb{R}$ ?