1. The problem statement, all variables and given/known data Here is a link to the proof I am reading: https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf 2. Relevant equations 3. The attempt at a solution The proof to which I am referring can be found on pages 8-9. At the top of page 9, the author makes an assertion which I endeavored to account for, but have been unsuccessful. Here is the assertion: $$0 \le U(f) - L(f) \le U(f,P - L(f;P) < \epsilon$$ Specifically, I am referring to $$U(f) - L(f) \ge 0$$. Is this really true; how do they know it will always be zero or positive? I have tried to justify it, but have failed. Could someone possibly help me?