 #1
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I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 3: Integration ... and in particular I am focused on Section 3.1: The Riemann Integral ...
I need help with an aspect of the proof of Proposition 3.1.4 ...Proposition 3.1.4 and its proof read as follows:
View attachment 9456
View attachment 9457
In the above proof by John Conway we read the following:
" ... ... Since \(\displaystyle \epsilon\) was arbitrary we have that there can be only one number between L(f, Q) and U(f, Q) for every such refinement. ... ... "My question is as follows:
Can someone please explain what Conway means by saying that there can be only one number between \(\displaystyle L(f, Q)\) and \(\displaystyle U(f, Q)\) for every such refinement and, further explain why this is true ... and then, further yet, why exactly this number is \(\displaystyle \int_a^b f \) ... Help will be appreciated ...
Peter==========================================================================================The above proof refers to Proposition 3.1.2 so I am making available the statement of the proposition ... as follows:View attachment 9458It may help MHB readers to have access to the start of Section 3.1 preliminary to Proposition 3.1.4 ... so I am providing the same ... as follows:View attachment 9459
View attachment 9460
View attachment 9461
Hope that helps ...
Peter
I need help with an aspect of the proof of Proposition 3.1.4 ...Proposition 3.1.4 and its proof read as follows:
View attachment 9456
View attachment 9457
In the above proof by John Conway we read the following:
" ... ... Since \(\displaystyle \epsilon\) was arbitrary we have that there can be only one number between L(f, Q) and U(f, Q) for every such refinement. ... ... "My question is as follows:
Can someone please explain what Conway means by saying that there can be only one number between \(\displaystyle L(f, Q)\) and \(\displaystyle U(f, Q)\) for every such refinement and, further explain why this is true ... and then, further yet, why exactly this number is \(\displaystyle \int_a^b f \) ... Help will be appreciated ...
Peter==========================================================================================The above proof refers to Proposition 3.1.2 so I am making available the statement of the proposition ... as follows:View attachment 9458It may help MHB readers to have access to the start of Section 3.1 preliminary to Proposition 3.1.4 ... so I am providing the same ... as follows:View attachment 9459
View attachment 9460
View attachment 9461
Hope that helps ...
Peter
Attachments

Conway  1  Proposition 3.1.4  PART 1 ... .png8.5 KB · Views: 86

Conway  2  Proposition 3.1.4  PART 2 ... .png26.9 KB · Views: 107

Conway  Proposition 3.1.2 ... .png10.3 KB · Views: 91

Conway  1  Start of Section 3.1 ... Part 1 .png31.5 KB · Views: 99

Conway  2  Start of Section 3.1 ... Part 2 ... .png60.1 KB · Views: 107

Conway  3  Start of Section 3.1 ... Part 3 ... .png42.1 KB · Views: 102
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