The Riemann Integral .... Conway, Proposition 3.1.4 ....

In summary, John B. Conway's book: A First Course in Analysis introduces the Riemann Integral. Chapter 3: Integration discusses the Riemann Integral and provides a proof of Proposition 3.1.4. My question is as follows: Can someone please explain what Conway means by saying that there can be only one number between L(f, Q) and U(f, Q) for every such refinement and, further explain why this is true ... and then, further yet, why exactly this number is \int_a^b f ... The wording is a little ambiguous. I can see how it could be interpreted as "for each refinement" and perhaps that is how you are reading it.
  • #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
 

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  • #2
That's a lot to digest in one sitting but here is your basic question:
Peter said:
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 \(\displaystyle \int_a^b f \) ...
The wording is a little ambiguous. I can see how it could be interpreted as "for each refinement" and perhaps that is how you are reading it. But he intends to say that there is a unique number that works for every refinement. Suppose there were two such numbers, p and q. Then then take a refinement such that [tex]\epsilon< |p- q|[/tex]. Those two numbers cannot both be in that refinement so cannot be in every refinement.

As for "why exactly this number is [tex]\int_a^b f [/tex]". He is defining [tex]\int_a^b f[/tex] to be that number!
 
  • #3
HallsofIvy said:
That's a lot to digest in one sitting but here is your basic question:

The wording is a little ambiguous. I can see how it could be interpreted as "for each refinement" and perhaps that is how you are reading it. But he intends to say that there is a unique number that works for every refinement. Suppose there were two such numbers, p and q. Then then take a refinement such that [tex]\epsilon< |p- q|[/tex]. Those two numbers cannot both be in that refinement so cannot be in every refinement.

As for "why exactly this number is [tex]\int_a^b f [/tex]". He is defining [tex]\int_a^b f[/tex] to be that number!
Thanks HallsofIvy ...

Still reflecting on what you have said ...

Thanks again ...

Peter
 

FAQ: The Riemann Integral .... Conway, Proposition 3.1.4 ....

1. What is the Riemann Integral?

The Riemann Integral is a mathematical concept used to calculate the area under a curve. It was developed by the mathematician Bernhard Riemann in the 19th century and is an important tool in calculus and analysis.

2. How is the Riemann Integral calculated?

The Riemann Integral is calculated by dividing the area under a curve into smaller rectangles and adding up the areas of these rectangles. As the width of the rectangles approaches zero, the sum of their areas approaches the exact area under the curve.

3. What is Proposition 3.1.4 in Conway's work on the Riemann Integral?

Proposition 3.1.4 is a theorem in John Conway's book "Functions of One Complex Variable I" that states that if a function is continuous on a closed interval, then it is also integrable on that interval. This proposition is important in proving the existence of the Riemann Integral for certain functions.

4. What is the significance of the Riemann Integral in mathematics?

The Riemann Integral is significant in mathematics because it allows for the calculation of areas and volumes of irregular shapes, which has many practical applications in fields such as physics, engineering, and economics. It is also a fundamental concept in calculus and analysis, playing a crucial role in the development of these branches of mathematics.

5. Are there other types of integrals besides the Riemann Integral?

Yes, there are other types of integrals, such as the Lebesgue Integral and the Cauchy Principal Value Integral. These integrals have different definitions and properties, but they all aim to calculate the area under a curve or the sum of a function over a given interval.

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