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I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...
I am focused on Chapter 7: The Riemann Integral ...
I need help in fully understanding an aspect of the proof of Theorem 7.2.9 ...Theorem 7.2.9 and its proof ... ... read as follows:View attachment 7319
https://www.physicsforums.com/attachments/7320In the above proof (near to the start ...) we read the following:
" ... let $$\dot{ \mathcal{P} }$$ be a tagged partition of $$[a, b]$$ with $$ \lvert \lvert \dot{ \mathcal{Q} } \lvert \lvert \ \lt \delta $$. ... ... "I am somewhat puzzled by B&S's use of $$\dot{ \mathcal{P} }$$ and $$\dot{ \mathcal{Q} }$$ in this proof ... can someone please explain the use of these symbols ... I know they are different partitions ... but why does B&S introduce them ... what is the logic ... why do we need both ... ... [ ... the use of both $$\dot{ \mathcal{P} }$$ and $$\dot{ \mathcal{Q} }$$ in the particular statement I quoted seems to me to be most peculiar ... ]
PeterIt may help readers of the above post to have reference to the notation of B&S in setting up the Riemann Integral ... so I am providing the text of Section 7.1 up to and including the definition of the Riemann Integral ... as follows ...View attachment 7321
https://www.physicsforums.com/attachments/7322
https://www.physicsforums.com/attachments/7323
I am focused on Chapter 7: The Riemann Integral ...
I need help in fully understanding an aspect of the proof of Theorem 7.2.9 ...Theorem 7.2.9 and its proof ... ... read as follows:View attachment 7319
https://www.physicsforums.com/attachments/7320In the above proof (near to the start ...) we read the following:
" ... let $$\dot{ \mathcal{P} }$$ be a tagged partition of $$[a, b]$$ with $$ \lvert \lvert \dot{ \mathcal{Q} } \lvert \lvert \ \lt \delta $$. ... ... "I am somewhat puzzled by B&S's use of $$\dot{ \mathcal{P} }$$ and $$\dot{ \mathcal{Q} }$$ in this proof ... can someone please explain the use of these symbols ... I know they are different partitions ... but why does B&S introduce them ... what is the logic ... why do we need both ... ... [ ... the use of both $$\dot{ \mathcal{P} }$$ and $$\dot{ \mathcal{Q} }$$ in the particular statement I quoted seems to me to be most peculiar ... ]
PeterIt may help readers of the above post to have reference to the notation of B&S in setting up the Riemann Integral ... so I am providing the text of Section 7.1 up to and including the definition of the Riemann Integral ... as follows ...View attachment 7321
https://www.physicsforums.com/attachments/7322
https://www.physicsforums.com/attachments/7323
Last edited: