MHB 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 $$\epsilon$$ 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 $$L(f, Q)$$ and $$U(f, Q)$$ for every such refinement and, further explain why this is true ... and then, further yet, why exactly this number is $$\int_a^b f $$ ... 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 $$L(f, Q)$$ 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. 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