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- Jun 22, 2012

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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:

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:

\(\displaystyle \int_a^b f \text{ exists } \Longrightarrow\) ... for every \(\displaystyle \epsilon \gt 0 \ \exists \ \) a partition \(\displaystyle \pi\) of \(\displaystyle [a, b]\) such that \(\displaystyle \overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon\) ... ...

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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:

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:

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:

Hope that helps ...

Peter

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