# The Riemann and Darboux Integrals ... Browder, Theorem 5.10 ... ...

#### 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 5.1 Riemann Sums ... ...

I need some help in understanding the proof of Theorem 5.10 ...

Theorem 5.10 and its proof read as follows:  At the start of the above proof by Browder we read the following:

" ... ... The necessity of the condition is immediate from the definition of the integral ... ... "

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Note: I am assuming that proving "the necessity of the condition is proving the following:

$$\displaystyle \int_a^b f \text{ exists } \Longrightarrow$$ ... for every $$\displaystyle \epsilon \gt 0 \ \exists \$$ a partition $$\displaystyle \pi$$ of $$\displaystyle [a, b]$$ such that $$\displaystyle \overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon$$ ... ...

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Help will be much appreciated ...

Peter

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Note: It may help MHB readers of the above post to have access to Browder's notation, definitions and theorems on Riemann integration preliminary to Theorem 5.10 ... hence i am providing access to the same ... as follows:    Hope that helps ...

Peter

Last edited:

#### 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 5.1 Riemann Sums ... ...

I need some help in understanding the proof of Theorem 5.10 ...

Theorem 5.10 and its proof read as follows:

At the start of the above proof by Browder we read the following:

" ... ... The necessity of the condition is immediate from the definition of the integral ... ... "

-------------------------------------------------------------------------------------------------------------------

Note: I am assuming that proving "the necessity of the condition is proving the following:

$$\displaystyle \int_a^b f \text{ exists } \Longrightarrow$$ ... for every $$\displaystyle \epsilon \gt 0 \ \exists \$$ a partition $$\displaystyle \pi$$ of $$\displaystyle [a, b]$$ such that $$\displaystyle \overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon$$ ... ...

-------------------------------------------------------------------------------------------------------------------

Help will be much appreciated ...

Peter

==========================================================================================

Note: It may help MHB readers of the above post to have access to Browder's notation, definitions and theorems on Riemann integration preliminary to Theorem 5.10 ... hence i am providing access to the same ... as follows:

Hope that helps ...

Peter

I have been reflecting on my problem in the above post and now give my attempted proof of

$$\displaystyle \int_a^b f \text{ exists } \Longrightarrow$$ ... for every $$\displaystyle \epsilon \gt 0 \ \exists \$$ a partition $$\displaystyle \pi$$ of $$\displaystyle [a, b]$$ such that $$\displaystyle \overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon$$ ... ...

Proof:

Let $$\displaystyle \int_a^b f = I$$

Then

$$\displaystyle I$$ exists $$\displaystyle \Longrightarrow$$ for any $$\displaystyle \frac{ \epsilon }{2} \gt 0 \ \exists \ \pi_0$$ such that for any $$\displaystyle \pi \geq \pi_0$$ and every selection $$\displaystyle \sigma$$ associated with $$\displaystyle \pi$$ we have $$\displaystyle | s(f, \pi, \sigma ) - I | \lt \frac{ \epsilon }{2}$$

Now $$\displaystyle | S(f, \pi, \sigma ) - I | \lt \frac{ \epsilon }{2}$$

implies that

$$\displaystyle - \frac{ \epsilon }{2} \lt S(f, \pi, \sigma ) - I \lt \frac{ \epsilon }{2}$$

and so, obviously, we have that

$$\displaystyle S(f, \pi, \sigma ) - I \lt \frac{ \epsilon }{2}$$ ... ... ... ... ... (1)

But $$\displaystyle | S(f, \pi, \sigma ) - I | \lt \frac{ \epsilon }{2}$$

... also implies that

$$\displaystyle - \frac{ \epsilon }{2} \lt I - S(f, \pi, \sigma ) \lt \frac{ \epsilon }{2}$$

so, obviously, we have that

$$\displaystyle I - S(f, \pi, \sigma ) \lt \frac{ \epsilon }{2}$$ ... ... ... ... ... (2)

Now we also have that

$$\displaystyle \underline{S} (f, \pi) \leq S(f, \pi, \sigma ) \leq \overline{S} (f, \pi)$$ ... ... ... ... ... (3)

Now (1) and (3) imply

$$\displaystyle \overline{S} (f, \pi) - I \lt \frac{ \epsilon }{2}$$ ... ... ... ... ... (4)

Similarly (2) and (3) imply

$$\displaystyle I - \underline{S} (f, \pi) \lt \frac{ \epsilon }{2}$$ ... ... ... ... ... (5)

$$\displaystyle \overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon$$ ... ...