MHB How can we prove the inequality for the supremum and infimum of f*g and f*g?

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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:
D&K ... Defn 6.2.7 and Theorem 6.2.8 .png

D&K ... Defn 6.2.7 and Theorem 6.2.8  ... PART II .png
The definition of supremum and infimum are given in the following text from D&K Vol. I ...
D&K ... Theorem 1.6.1 including Defn of Sup A .png

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 [math]f: X \to \mathbb{R}[/math] is a real-valued function and [math] A \subset X [/math] ... ... ...

... then we define ..

[math] \text{ sup}_B = \text{sup} \{ f(x) \ | \ x \in B \} [/math]

and

[math] \text{ inf}_B = \text{inf} \{ f(x) \ | \ x \in B \} [/math]
I need help to formulate a detailed, formal and rigorous proof that [math] \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 [/math]I have been unable to make a meaningful start on this proof ...Help will be much appreciated ...

Peter
 
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For every $x\in B$, $f(x) \leqslant \sup_B f$ and $g(x) \leqslant \sup_B g$. Therefore $(fg)(x) = f(x)g(x) \leqslant \sup_B f \sup_B g$. Now take the sup over $B$ to get $\sup_B fg \leqslant \sup_B f \sup_B g$. A similar argument shows that $\inf_B fg \geqslant \inf_B f \inf_B g$ and so $-\inf_B fg \leqslant -\inf_B f \inf_B g$.
 
Thanks Opalg …

… very much appreciate your help …

Just reflecting on what you have written …

Peter
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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