The integral of f on [a,b] exists and is positive.
Prove there is a subinterval J of [a,b] and a constant c such that f(x) >= c > 0 for all x in J.
Hint: Consider the lower integral of f on [a,b]
The Attempt at a Solution
I don't see how the hint helps. Obviously both the upper and lower integrals are greater than 0. I tried considering refinement partitions of J extending to [a,b] but that doesn't get me back to getting the actual function bounded.
Since the lower integral is greater than 0, the lower sum for any partition is greater than 0 as well.... I just don't see how this is helping and how to go from the sum/integral back to the original function. Is there a different approach that could be taken? Any help? Thx :)