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I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ...

I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof reads as follows:

The definition of supremum and infimum are given in the following text from D&K Vol. I ...

I cannot locate D&K's definition of sup and inf for functions so I am taking the definition from Joseph L. Taylor's book, "Foundations of Analysis".

Taylor's definition reads as follows:

If \(\displaystyle f: X \to \mathbb{R}\) is a real-valued function and \(\displaystyle A \subset X \) ... ... ...

... then we define ..

\(\displaystyle \text{ sup}_B = \text{sup} \{ f(x) \ | \ x \in B \} \)

and

\(\displaystyle \text{ inf}_B = \text{inf} \{ f(x) \ | \ x \in B \} \)

Peter

I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof reads as follows:

I cannot locate D&K's definition of sup and inf for functions so I am taking the definition from Joseph L. Taylor's book, "Foundations of Analysis".

Taylor's definition reads as follows:

If \(\displaystyle f: X \to \mathbb{R}\) is a real-valued function and \(\displaystyle A \subset X \) ... ... ...

... then we define ..

\(\displaystyle \text{ sup}_B = \text{sup} \{ f(x) \ | \ x \in B \} \)

and

\(\displaystyle \text{ inf}_B = \text{inf} \{ f(x) \ | \ x \in B \} \)

**\(\displaystyle \text{ sup}_B \ fg - \text{ inf}_B \ fg \leq \text{ sup}_B \ f \text{ sup}_B \ g \ - \ \text{ inf}_B \ f \text{ inf}_B \ g \)I have been unable to make a meaningful start on this proof ...Help will be much appreciated ...**__I need help to formulate a detailed, formal and rigorous proof that__Peter

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