- #1
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- TL;DR Summary
- I need help to prove a theorem effectively equivalencing Riemann and Darboux integration ...
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:
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 ...
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Note: I am assuming that proving "the necessity of the condition is proving the following:##\int_a^b f \text{ exists } \Longrightarrow## ... for every ##\epsilon \gt 0 \ \exists \ ## a partition ##\pi## of ##[a, b]## such that ##\overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon## ... ...
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Help will be much appreciated ...
Peter
==========================================================================================Note: It may help Physics Forum 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:
Hope that text 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:
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:##\int_a^b f \text{ exists } \Longrightarrow## ... for every ##\epsilon \gt 0 \ \exists \ ## a partition ##\pi## of ##[a, b]## such that ##\overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon## ... ...
-------------------------------------------------------------------------------------------------------------------
Help will be much appreciated ...
Peter
==========================================================================================Note: It may help Physics Forum 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:
Peter