Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2

[Math Amateur]

I am reading Manfred Stoll's book: Introduction to Real Analysis.

I need help with Stoll's proof of The Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2

Stoll's statement of Theorem 6.3.2 and its proof reads as follows:View attachment 3952
View attachment 3953
In the above proof we read:

Since

\(\displaystyle \mathscr{L}( \mathscr{P} , f ) \le \sum_{i = 1}^n f(t_i) \Delta x_i \le \mathscr{U}( \mathscr{P} , f )\)

we have

\(\displaystyle \underline{\int^b_a}f(x) dx \le F(b) - F(a) \le \overline{\int^b_a} f(x) dx \)My question is as follows: How do we know this is true? or Why exactly is this the case?

Can someone explain?
To restate my question:Since \(\displaystyle \underline{\int^b_a} f(x) dx\) is the supremum of \(\displaystyle \mathscr{L}( \mathscr{P} , f )\) over all partitions \(\displaystyle \mathscr{P}\) how do we know that it is less than F(b) - F(a)? ... ...... ... that is, is it possible for the quantity \(\displaystyle \mathscr{L}( \mathscr{P} , f ) \)to be less than \(\displaystyle F(b) - F(a)\) for all partitions \(\displaystyle \mathscr{P}\) but for the supremum over all partitions to fail to be less than \(\displaystyle F(b) - F(a)\)? ... ...
Hmm ... ... reflecting ... ... beginning to think this is not possible ... but how would you frame a rigorous symbolic argument of this?

Can someone please help?

Peter

 

[Math Amateur]

I am reading Manfred Stoll's book: Introduction to Real Analysis.

I need help with Stoll's proof of The Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2

Stoll's statement of Theorem 6.3.2 and its proof reads as follows:

In the above proof we read:

Since

\(\displaystyle \mathscr{L}( \mathscr{P} , f ) \le \sum_{i = 1}^n f(t_i) \Delta x_i \le \mathscr{U}( \mathscr{P} , f )\)

we have

\(\displaystyle \underline{\int^b_a}f(x) dx \le F(b) - F(a) \le \overline{\int^b_a} f(x) dx \)My question is as follows: How do we know this is true? or Why exactly is this the case?

Can someone explain?
To restate my question:Since \(\displaystyle \underline{\int^b_a} f(x) dx\) is the supremum of \(\displaystyle \mathscr{L}( \mathscr{P} , f )\) over all partitions \(\displaystyle \mathscr{P}\) how do we know that it is less than F(b) - F(a)? ... ...... ... that is, is it possible for the quantity \(\displaystyle \mathscr{L}( \mathscr{P} , f ) \)to be less than \(\displaystyle F(b) - F(a)\) for all partitions \(\displaystyle \mathscr{P}\) but for the supremum over all partitions to fail to be less than \(\displaystyle F(b) - F(a)\)? ... ...
Hmm ... ... reflecting ... ... beginning to think this is not possible ... but how would you frame a rigorous symbolic argument of this?

Can someone please help?

Peter

After a little reflection, I will try to answer my own question ...We have that:

\(\displaystyle \mathscr{L}( \mathscr{P} , f ) \le F(b) - F(a)\) for any (i.e. every) partition \(\displaystyle \mathscr{P}\) ... ... ... (1)
We want to show that it follows from (1) that:\(\displaystyle \text{ sup } \{ \mathscr{L}( \mathscr{P} , f ) \} \le F(b) - F(a) \) ... ... ... (2)
Now proceed with proof:Suppose (2) does not follow ... ... then we have:\(\displaystyle \text{ sup } \{ \mathscr{L}( \mathscr{P} , f ) \} \gt F(b) - F(a)\) ... ... ... (3)Then there exists an \(\displaystyle \epsilon \gt 0\) such that:

\(\displaystyle sup \{ \mathscr{L}( \mathscr{P} , f ) \} - \epsilon = F(b) - F(a)\) ... ... ... (4)Then we have that:

\(\displaystyle \text{ sup } \{ \mathscr{L}( \mathscr{P} , f ) \} - \frac{\epsilon}{2} \gt F(b) - F(a)\) ... ... ... (5)But from Lemma 1.2.10 in Pons: Real Analysis for Undergraduates (see below and also see my reply to Fallen Angel in my post:http://mathhelpboards.com/analysis-50/riemann-criterion-integrability-stoll-theorem-6-17-a-14292.html ) we have that there is a partition \(\displaystyle \mathscr{P}_{ \frac{\epsilon}{2} }\) such that:\(\displaystyle \text{ sup } \{ \mathscr{L}( \mathscr{P} , f ) \} - \frac{\epsilon}{2} \lt \mathscr{L} ( \mathscr{P}_{ \frac{\epsilon}{2} } , f ) \) ... ... ... (6)
Now (5), (6) imply that:\(\displaystyle \mathscr{L} ( \mathscr{P}_{ \frac{\epsilon}{2} } , f ) \gt F(b) - F(a) \)

which is a contradiction ... ...
I am somewhat unsure of the correctness and reasonableness of my analysis ... ... so ... Can someone please critique the above analysis, pointing out any errors or misconceptions ...

Would appreciate some help ... ...

Peter***EDIT***

In the above post I mentioned Pons' Lemma 1.2.10 which I referred to and provided in a previous post.

For the convenience of MHB members reading this post I have decided to re-provide it here, as follows:
View attachment 3958
https://www.physicsforums.com/attachments/3959

 

[Fallen Angel]

Hi Peter,

Your argument is correct but there is a direct way of doing this.

$\mathcal{L}(\mathcal{P},f)\leq F(a)-F(b)$ for any partition $\mathcal{P}$ means that $F(a)-F(b)$ is an upper bound for $\{\mathcal{L}(\mathcal{P},f)\}$ and the supremum is always less or equal than any upper bound.