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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 \(\displaystyle f\) is integrable on \(\displaystyle [a,b]\). Let \(\displaystyle \epsilon \gt 0\) be given. Then there exist partitions \(\displaystyle \mathscr{P}_1\) and \(\displaystyle \mathscr{P}_2\) of \(\displaystyle [a,b]\) such that
\(\displaystyle \mathscr{U}( \mathscr{P}_2 , f ) - \int^b_a f \lt \frac{ \epsilon }{2}\) ... ... ... (3)\(\displaystyle \int^b_a f - \mathscr{L}( \mathscr{P}_1 , f ) \lt \frac{ \epsilon }{2}\) ... ... ... (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
\(\displaystyle \overline{\int^b_a}f = \text{inf} \{ \mathscr{U}( \mathscr{P} , f ): \mathscr{P} \text{ is a partition of } [a,b] \}\)
and
\(\displaystyle \underline{\int^b_a}f = \text{sup} \{ \mathscr{L}( \mathscr{P} , f ): \mathscr{P} \text{ is a partition of } [a,b] \}\)
and, further that
\(\displaystyle \int^b_a f = \underline{\int^b_a} f = \overline{\int^b_a} f
\)
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
View attachment 3944
View attachment 3945
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 \(\displaystyle f\) is integrable on \(\displaystyle [a,b]\). Let \(\displaystyle \epsilon \gt 0\) be given. Then there exist partitions \(\displaystyle \mathscr{P}_1\) and \(\displaystyle \mathscr{P}_2\) of \(\displaystyle [a,b]\) such that
\(\displaystyle \mathscr{U}( \mathscr{P}_2 , f ) - \int^b_a f \lt \frac{ \epsilon }{2}\) ... ... ... (3)\(\displaystyle \int^b_a f - \mathscr{L}( \mathscr{P}_1 , f ) \lt \frac{ \epsilon }{2}\) ... ... ... (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
\(\displaystyle \overline{\int^b_a}f = \text{inf} \{ \mathscr{U}( \mathscr{P} , f ): \mathscr{P} \text{ is a partition of } [a,b] \}\)
and
\(\displaystyle \underline{\int^b_a}f = \text{sup} \{ \mathscr{L}( \mathscr{P} , f ): \mathscr{P} \text{ is a partition of } [a,b] \}\)
and, further that
\(\displaystyle \int^b_a f = \underline{\int^b_a} f = \overline{\int^b_a} f
\)
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
View attachment 3944
View attachment 3945
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