Welcome to our community

Be a part of something great, join today!

Riemann Integration .. Existence Result ... Browder, Theorem 5.12 ...

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 5.2 Existence Results ... ...

I need some help in understanding the proof of Theorem 5.12 ...


Theorem 5.12 and its proof read as follows:



Browder ... Theorem 5.12 .png


In the above proof by Andrew Browder we read the following:

" ... ... [For instance, one can choose a positive integer \(\displaystyle n\) such that \(\displaystyle n \gt [f(b) - f(a) + 1](b - a) / \epsilon\) ... ... "


My question is as follows:

Why does Browder have \(\displaystyle +1\) in the expression \(\displaystyle [f(b) - f(a) + 1](b - a) / \epsilon\) ... ... ?


Surely \(\displaystyle [f(b) - f(a)](b - a) / \epsilon\) will do fine ... since ...

\(\displaystyle \mu ( \pi ) = (b - a)/ n\)

and so

\(\displaystyle \mu ( \pi ) [f(b) - f(a)] = [f(b) - f(a)] (b - a)/ n \lt \epsilon\) ...

... so we only need ...

\(\displaystyle n \gt [f(b) - f(a)](b - a) / \epsilon\)




Hope someone can help ...

Peter