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 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 where 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.