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The Riemann and Darboux Integrals ... Browder, Theorem 5.10 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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:


Browder - 1 - Theorem 5.10 ... PART 1 ... .png
Browder - 2 - Theorem 5.10 ... PART 2 ... .png




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:



Browder - 1 - Start of 5.1 - Relevant Defns & Propns ... PART 1 ... .png
Browder - 2 - Start of 5.1 - Relevant Defns & Propns ... PART 2 ... .png
Browder - 3 - Start of 5.1 - Relevant Defns & Propns ... PART 3 ... .png
Browder - 4 - Start of 5.1 - Relevant Defns & Propns ... PART 4 ... .png



Hope that helps ...

Peter
 
Last edited:

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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 ...

-------------------------------------------------------------------------------------------------------------------

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




I have been reflecting on my problem in the above post and now give my attempted proof of

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


Proof:

Let \(\displaystyle \int_a^b f = I \)

Then

\(\displaystyle I\) exists \(\displaystyle \Longrightarrow\) for any \(\displaystyle \frac{ \epsilon }{2} \gt 0 \ \exists \ \pi_0\) such that for any \(\displaystyle \pi \geq \pi_0\) and every selection \(\displaystyle \sigma\) associated with \(\displaystyle \pi\) we have \(\displaystyle | s(f, \pi, \sigma ) - I | \lt \frac{ \epsilon }{2}\)


Now \(\displaystyle | S(f, \pi, \sigma ) - I | \lt \frac{ \epsilon }{2}\)

implies that

\(\displaystyle - \frac{ \epsilon }{2} \lt S(f, \pi, \sigma ) - I \lt \frac{ \epsilon }{2}\)

and so, obviously, we have that

\(\displaystyle S(f, \pi, \sigma ) - I \lt \frac{ \epsilon }{2}\) ... ... ... ... ... (1)


But \(\displaystyle | S(f, \pi, \sigma ) - I | \lt \frac{ \epsilon }{2}\)

... also implies that

\(\displaystyle - \frac{ \epsilon }{2} \lt I - S(f, \pi, \sigma ) \lt \frac{ \epsilon }{2}\)

so, obviously, we have that

\(\displaystyle I - S(f, \pi, \sigma ) \lt \frac{ \epsilon }{2}\) ... ... ... ... ... (2)

Now we also have that

\(\displaystyle \underline{S} (f, \pi) \leq S(f, \pi, \sigma ) \leq \overline{S} (f, \pi) \) ... ... ... ... ... (3)


Now (1) and (3) imply


\(\displaystyle \overline{S} (f, \pi) - I \lt \frac{ \epsilon }{2}\) ... ... ... ... ... (4)


Similarly (2) and (3) imply


\(\displaystyle I - \underline{S} (f, \pi) \lt \frac{ \epsilon }{2}\) ... ... ... ... ... (5)


Adding (4) and (5) gives


\(\displaystyle \overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon\) ... ...



Can someone please critique the above proof and either confirm it is correct and/or point out errors or shortcomings ...

Peter
 
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