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

#### Peter

##### Well-known member
MHB Site Helper
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: 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