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- Jun 22, 2012

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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:

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:

It 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:

Hope that helps ...

Peter

I need help with an aspect of the proof of Proposition 3.1.4 ...

Proposition 3.1.4 and its proof read as follows:

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:

It 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:

Hope that helps ...

Peter

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