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Riemann's Integrability Condition

  1. Mar 29, 2015 #1
    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?
  2. jcsd
  3. Mar 29, 2015 #2


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    Am I misunderstanding, or did you happen to oversee proposition 1.13 and its proof, just above section 1.4?
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