Riemann's Integrability Condition

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SUMMARY

The discussion centers on Riemann's Integrability Condition, specifically the assertion that the upper sum \( U(f) \) minus the lower sum \( L(f) \) is always non-negative, as stated in the proof found on pages 8-9 of the referenced document. The participant seeks clarification on the validity of the inequality \( 0 \le U(f) - L(f) \) and its implications for integrability. The confusion arises from the interpretation of the proof, particularly regarding proposition 1.13 and its relevance to the condition being discussed.

PREREQUISITES
  • Understanding of Riemann sums and their definitions.
  • Familiarity with the concepts of upper and lower sums in integration.
  • Knowledge of mathematical proofs and inequalities.
  • Basic understanding of the properties of integrable functions.
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  • Review the proof of Riemann's Integrability Condition in detail.
  • Study the implications of proposition 1.13 in the context of Riemann integration.
  • Explore examples of functions that illustrate the upper and lower sums.
  • Learn about the criteria for a function to be Riemann integrable.
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Bashyboy
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Homework Statement


Here is a link to the proof I am reading: https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf

Homework Equations

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?
 
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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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