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

[Math Amateur]

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:
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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
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Hope that helps ...

Peter

 

[HallsofIvy]

That's a lot to digest in one sitting but here is your basic question:​

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 $$\epsilon< |p- q|$$. Those two numbers cannot both be in that refinement so cannot be in every refinement.

As for "why exactly this number is $$\int_a^b f$$". He is defining $$\int_a^b f$$ to be that number!

 

[Math Amateur]

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 $$\epsilon< |p- q|$$. Those two numbers cannot both be in that refinement so cannot be in every refinement.

As for "why exactly this number is $$\int_a^b f$$". He is defining $$\int_a^b f$$ to be that number!

Thanks HallsofIvy ...

Still reflecting on what you have said ...

Thanks again ...

Peter