Riemann's Integrability Condition

[Bashyboy]

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?

 

[BvU]

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