 #1
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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ...
I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof read as follows:View attachment 9473
View attachment 9474
At the start of the above proof by Browder we read the following:
" ... ... The necessity of the condition is immediate from the definition of the integral ... ... "Can someone please help me to rigorously demonstrate the necessity of the condition ...

Note: I am assuming that proving "the necessity of the condition is proving the following:\(\displaystyle \int_a^b f \text{ exists } \Longrightarrow\) ... for every \(\displaystyle \epsilon \gt 0 \ \exists \ \) a partition \(\displaystyle \pi\) of \(\displaystyle [a, b]\) such that \(\displaystyle \overline{S} (f, \pi)  \underline{S} (f, \pi) \lt \epsilon\) ... ...

Help will be much appreciated ...
Peter
==========================================================================================Note: It may help MHB readers of the above post to have access to Browder's notation, definitions and theorems on Riemann integration preliminary to Theorem 5.10 ... hence i am providing access to the same ... as follows:
View attachment 9475
View attachment 9476
View attachment 9477
View attachment 9478
Hope that helps ...
Peter
I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ...
I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof read as follows:View attachment 9473
View attachment 9474
At the start of the above proof by Browder we read the following:
" ... ... The necessity of the condition is immediate from the definition of the integral ... ... "Can someone please help me to rigorously demonstrate the necessity of the condition ...

Note: I am assuming that proving "the necessity of the condition is proving the following:\(\displaystyle \int_a^b f \text{ exists } \Longrightarrow\) ... for every \(\displaystyle \epsilon \gt 0 \ \exists \ \) a partition \(\displaystyle \pi\) of \(\displaystyle [a, b]\) such that \(\displaystyle \overline{S} (f, \pi)  \underline{S} (f, \pi) \lt \epsilon\) ... ...

Help will be much appreciated ...
Peter
==========================================================================================Note: It may help MHB readers of the above post to have access to Browder's notation, definitions and theorems on Riemann integration preliminary to Theorem 5.10 ... hence i am providing access to the same ... as follows:
View attachment 9475
View attachment 9476
View attachment 9477
View attachment 9478
Hope that helps ...
Peter
Attachments

Browder  1  Theorem 5.10 ... PART 1 ... .png22.9 KB · Views: 95

Browder  2  Theorem 5.10 ... PART 2 ... .png4.8 KB · Views: 87

Browder  1  Start of 5.1  Relevant Defns & Propns ... PART 1 ... .png18.1 KB · Views: 96

Browder  2  Start of 5.1  Relevant Defns & Propns ... PART 2 ... .png57.5 KB · Views: 99

Browder  3  Start of 5.1  Relevant Defns & Propns ... PART 3 ... .png44.6 KB · Views: 88

Browder  4  Start of 5.1  Relevant Defns & Propns ... PART 4 ... .png43.2 KB · Views: 89
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