Fundamental Theorem Of Calculus (Second Form) - B&S Theorem 7.3.5 .... ....

[Math Amateur]

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...

I am focused on Chapter 7: The Riemann Integral ...

I need help in fully understanding an aspect of the proof of Theorem 7.3.5 ...Theorem 7.3.5 and its proof ... ... read as follows:
View attachment 7325In the above proof from Bartle and Sherbert we read the following:

" ... ... Now on the interval $$[c, c + h]$$ the function $$f$$ satisfies inequality (4), so that we have

$$( f(c) - \epsilon ) \cdot h \le F( c + h ) - F(c) = \int^{ c + h }_c f \le ( f(c) + \epsilon ) \cdot h$$

... ... "Can someone please demonstrate rigorously and in detail how Bartle and Sherbert arrived at

$$( f(c) - \epsilon ) \cdot h \le F( c + h ) - F(c) = \int^{ c + h }_c f \le ( f(c) + \epsilon ) \cdot h$$ ... ... ?Peter================================================================================

It may help readers of the above post to have access to B&S's definition of the indefinite integral of $$f$$ ... ... so I am providing the same ... ... as follows:View attachment 7326

 

[Math Amateur]

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...

I am focused on Chapter 7: The Riemann Integral ...

I need help in fully understanding an aspect of the proof of Theorem 7.3.5 ...Theorem 7.3.5 and its proof ... ... read as follows:
In the above proof from Bartle and Sherbert we read the following:

" ... ... Now on the interval $$[c, c + h]$$ the function $$f$$ satisfies inequality (4), so that we have

$$( f(c) - \epsilon ) \cdot h \le F( c + h ) - F(c) = \int^{ c + h }_c f \le ( f(c) + \epsilon ) \cdot h$$

... ... "Can someone please demonstrate rigorously and in detail how Bartle and Sherbert arrived at

$$( f(c) - \epsilon ) \cdot h \le F( c + h ) - F(c) = \int^{ c + h }_c f \le ( f(c) + \epsilon ) \cdot h$$ ... ... ?Peter================================================================================

It may help readers of the above post to have access to B&S's definition of the indefinite integral of $$f$$ ... ... so I am providing the same ... ... as follows:

On reflection I think that the explanation for my question is as follows:

Since $$f(c) - \epsilon$$ is less than $$f(x)$$ for all $$x$$ in $$c \le x \lt c + h$$ ... ... ... we have that $$( f(c) - \epsilon ) \cdot h \le \int^{ c + h }_c f$$ ... ...Is that basically the correct explanation ... ... ?Peter

 

[Opalg]

On reflection I think that the explanation for my question is as follows:

Since $$f(c) - \epsilon$$ is less than $$f(x)$$ for all $$x$$ in $$c \le x \lt c + h$$ ... ... ... we have that $$( f(c) - \epsilon ) \cdot h \le \int^{ c + h }_c f$$ ... ...Is that basically the correct explanation ... ... ?

Yes. :)