Uniform Convergence and Integrals

**zebra2147** · March 8th 2011, 03:54 PM

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.

**roninpro** · March 8th 2011, 05:45 PM

Isn't the term on the right a Riemann sum using rectangles of uniform width $1/n$ ?

**zebra2147** · March 8th 2011, 06:08 PM

So then the height of the rectangle would be defined by $f(\frac{x}{n})$ since $f$ maps $[0,1]\rightarrow \mathbb{R}$ ?

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