MHB Riemann Criterion for Integrability - Stoll: Theorem 6.17

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I am reading Manfred Stoll's book: Introduction to Real Analysis.

I need help with Stoll's proof of Riemann's Criterion for Integrability - Stoll: Theorem 6.17

Stoll's statement of this theorem and its proof reads as follows:
https://www.physicsforums.com/attachments/3941In the above proof we read the following:

" ... ... Conversely, suppose $$f$$ is integrable on $$[a,b]$$. Let $$\epsilon \gt 0$$ be given. Then there exist partitions $$\mathscr{P}_1$$ and $$\mathscr{P}_2$$ of $$[a,b]$$ such that

$$\mathscr{U}( \mathscr{P}_2 , f ) - \int^b_a f \lt \frac{ \epsilon }{2}$$ ... ... ... (3)$$\int^b_a f - \mathscr{L}( \mathscr{P}_1 , f ) \lt \frac{ \epsilon }{2}$$ ... ... ... (4)

... ... "My problem is that I am concerned as to why exactly (3) (4) follow ... ...

(I know they seem plausible ... but what is a rigorous argument to say they hold ... )Presumably (3) (4) follow because

$$\overline{\int^b_a}f = \text{inf} \{ \mathscr{U}( \mathscr{P} , f ): \mathscr{P} \text{ is a partition of } [a,b] \}$$

and

$$\underline{\int^b_a}f = \text{sup} \{ \mathscr{L}( \mathscr{P} , f ): \mathscr{P} \text{ is a partition of } [a,b] \}$$

and, further that

$$\int^b_a f = \underline{\int^b_a} f = \overline{\int^b_a} f
$$

Is this correct?

However ... ... even if this is correct I am concerned to see an rigorous and exact explanation of why (3) and (4) above are true ...

Can someone please help?

Peter
So MHB members who are interested can follow the notation and definitions behind this post I am now providing the first basic elements of Stoll's presentation of the the theory of integration ... as follows ... ...
View attachment 3942
View attachment 3943
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Hi Peter,

It follows from that, if the Riemann integral exists, then both infimum and supremum must exist and be equal to the integral.

Now take a paper, write down the infimum definition and the statement you want to prove, move your head a little bit away and realize they are the same. ;)
 
Fallen Angel said:
Hi Peter,

It follows from that, if the Riemann integral exists, then both infimum and supremum must exist and be equal to the integral.

Now take a paper, write down the infimum definition and the statement you want to prove, move your head a little bit away and realize they are the same. ;)
Yes ... probably overthinking this issue ...

Thanks Fallen Angel ...

Peter
 
Peter said:
Yes ... probably overthinking this issue ...

Thanks Fallen Angel ...

Peter
Hi again Fallen Angel ...

Your post sent me searching my various texts for definitions and propositions concerning supremums and infimums to ensure I understod the notions and what they implied ...

One of the books was Matthew A. Pons book: Real Analysis for the Undergraduate ... and ... after his definition of a supremum we find Lemma 1.2.10 and some remarks on it and the proof ... ! .. just what I wanted ...

I am now studying the remarks and the proof ...

For the interests of MHB members, here is the definition followed by Lemma 1.2.10 and its proof ...
https://www.physicsforums.com/attachments/3946
View attachment 3947I think Lemma 1.2.10 thoroughly clarifies the issue ...

My admiration for Pons as a rigorous and clear text just went up a notch ...

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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