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

[Math Amateur]

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:
View attachment 9501In 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

 

[Jhoneine]

The +1 in the expression [f(b) - f(a) + 1](b - a) / \epsilon is used to ensure that the inequality holds in all cases. If we only use [f(b) - f(a)](b - a) / \epsilon, we can only guarantee that the inequality will hold if f(b) ≥ f(a). However, if f(b) < f(a) then we can't guarantee that the inequality will hold. The +1 ensures that the inequality holds for any value of f(b) and f(a).

 

[soloenergy]

Hi Peter,

I can definitely understand your confusion with the +1 in the expression. Here's my understanding of it:

In this proof, Browder is trying to show that for any given \epsilon \gt 0, there exists a partition \pi of [a,b] such that \mu(\pi)[f(b) - f(a)] \lt \epsilon. In order to do this, he chooses a positive integer n such that n \gt [f(b) - f(a) + 1](b - a) / \epsilon.

The reason for the +1 is because Browder wants to make sure that the value of n is large enough to guarantee that \mu(\pi)[f(b) - f(a)] is less than \epsilon. By adding 1 to [f(b) - f(a)], he is ensuring that n is large enough to cover any potential "gaps" in the interval [f(b) - f(a), \epsilon].

In other words, if we only have n \gt [f(b) - f(a)](b - a) / \epsilon, there is a possibility that the value of n may not be large enough to cover all possible values within the interval [f(b) - f(a), \epsilon]. By adding 1, we are ensuring that n is large enough to cover all possible values within this interval.

I hope this helps clarify the use of +1 in the expression. Let me know if you have any further questions.