Suppose that the bounded function f:[a,b]->R has the property that f(x)>=0 for a

Question

Suppose that the bounded function f:[a,b]->R has the property that f(x)>=0 for all x in [a,b]. Prove that the lower integral from a to b of f is greater than or equal to 0.

Explanation / Answer

Between two points of the partition P, the lower Riemann sum L(f, P) approximates the function by its infimum between those two points. Since there must be a rational number between them, that infimum will be less than or equal to 0. Each term in the lower sum is then negative or 0, so the whole thing has L(f, P) >= 0. Taking the supremum of the L(f, P)'s, it must be >= 0, so the lower Riemann integral is = 0. I have no idea what you meant by splitting it into cases. Your U(f, P) and L(f, P) equations also make no sense to me. The sup{0} = 0 and inf{0} = 0 are similarly mystifying--you realize a lower sum may well give a negative approximation of the area, and an upper sum may well give a positive one? These actually occur for the function f(x) = x if x is irrational and 0 if x is rational, where [a, b] = [-1, 1]. It seems like you don't really understand the concepts involved.

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