MHB Understand Proposition 1.3 in Conway's Functions of Complex Variables I

[Math Amateur]

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ...

I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ...

I need help in fully understanding aspects of Proposition 1.3 ...Proposition 1.3 and its proof read as follows:View attachment 7436
https://www.physicsforums.com/attachments/7437
https://www.physicsforums.com/attachments/7438In the above text from Palka, we read the following:

" ... ... Also, we may choose $$\delta_2 \gt 0$$ such that if $$P = \{ a = t_0 \lt t_1 \lt \ ... \ \lt t_m = b \}$$ and $$\lvert \lvert P \rvert \rvert = \text{ max } \{ (t_k - t_{ k - 1 } ) : \ 1 \le k \le m \} \lt \delta_2$$ then $$\left\lvert \int_a^b \lvert \gamma' (t) \rvert \ dt - \sum_{ k = 1 }^m \lvert \gamma' ( \tau_k ) \rvert ( t_k - t_{ k - 1 } ) \right\rvert \lt \epsilon$$

where $$\tau_k$$ is any point in $$[ t_{ k - 1 } , t_k ]$$. ... ... "Can someone please explain how/why the above quoted statement is true ... ...?

Is it simply a restatement of the condition for the Riemann integral to exist ... ?
Help will be much appreciated ... ...

Peter

 

[Opalg]

" ... ... Also, we may choose $$\delta_2 \gt 0$$ such that if $$P = \{ a = t_0 \lt t_1 \lt \ ... \ \lt t_m = b \}$$ and $$\lvert \lvert P \rvert \rvert = \text{ max } \{ (t_k - t_{ k - 1 } ) : \ 1 \le k \le m \} \lt \delta_2$$ then

$$\left\lvert \int_a^b \lvert \gamma' (t) \rvert \ dt - \sum_{ k = 1 }^m \lvert \gamma' ( \tau_k ) \rvert ( t_k - t_{ k - 1 } ) \right\rvert \lt \epsilon$$

where $$\tau_k$$ is any point in $$[ t_{ k - 1 } , t_k ]$$. ... ... "Can someone please explain how/why the above quoted statement is true ... ...?

Is it simply a restatement of the condition for the Riemann integral to exist ... ?

Yes!