Hi,

Ok so i have this for a homework problem:

Prove any non-negative Riemann integrable function is bounded.

So far in class we haven't learned much about Riemann sums excepts how to construct them. And we learned the properties of integrals, like $\displaystyle \int{Cf(x)}=C\int{f(x)}$ and other simple properties.

I wanted to try to prove this by contrapostive but I can't get it to work.

Here's my idea:

Assume f is non-negative and unbounded on [a,b].

(We want to show f is not Riemann integrable)

Consider the Riemann sum obtained by splitting [a,b] into n subintervals of length 1/n.

Then $\displaystyle R=\sum{f(x_i)\frac{1}{n}}$ where $\displaystyle x_i$ is an arbitrary point in a single interval of the partition of [a,b]

Now I want to use the fact the f is not bounded to show the limit as n goes to infinity doesn't exist.

But since 1/n goes to 0 I'm not sure how to proceed.

PLEASE any help or suggestions would be greatly appreciated.