Understanding Cantor's Theorem and Diameter

[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 an aspect of the proof of Theorem 1.4 ... ...

Theorem 1.4 and its proof read as follows:View attachment 7443
View attachment 7444In the above text from Conway we read the following:

" ... ... Let us show that this will complete the proof. If \(\displaystyle \epsilon \gt 0\) let \(\displaystyle m \gt ( 2/ \epsilon ) V( \gamma ) \ge \text{ diam } F_m\). Since \(\displaystyle I \in F_m\) , \(\displaystyle F_m \subset B( I ; \epsilon )\). Thus if \(\displaystyle \delta = \delta_m\) the theorem is proved. ... ... "I confess I am a little lost here ...I can see that if \(\displaystyle \epsilon \gt 0\) then \(\displaystyle F_m \subset B( I ; \epsilon )\) ... but how exactly does this fact together with \(\displaystyle \delta = \delta_m\) assure us that the theorem is proved ...

... ... that is ... how does \(\displaystyle F_m \subset B( I ; \epsilon )\)

\(\displaystyle \Longrightarrow\) there exists a complex number \(\displaystyle I\) such that for every \(\displaystyle \epsilon \gt 0\) there exists a \(\displaystyle \delta \gt 0\) sch that for \(\displaystyle \lvert \lvert P \rvert \rvert \lt \delta\) ... then ...

\(\displaystyle \left\lvert I - \sum_{ i = 1}^m f( \tau_k ) [ \gamma (t_k) -\gamma ( t_{ k-1 } ) ] \right\rvert \)
Help will be much appreciated ...

Peter==================================================================================

In the above proof, Conway mentions Cantor's Theorem ... so I am providing MHB readers with Conway's statement of Cantor's Theorem together with the relevant definition of the diameter of a set ... as follows:

View attachment 7445

 

[Euge]

If $\delta = \delta_m$ and $P = \{t_0 < t_1 < \cdots < t_n\}$ is a partition of $[a,b]$ with $\|P\| < \delta$ and $\tau_k\in [t_{k-1},t_k]$ for all $k$, then $P \in \mathscr{P}_m$ and $\sum f(\tau_k) [\gamma(t_k) - \gamma(t_{k-1})] \in F_m\subset B(I, \epsilon)$, which implies $\lvert I - \sum f(\tau_k)[\gamma(t_k) - \gamma(t_{k-1})\rvert < \epsilon$.

 

[Math Amateur]

If $\delta = \delta_m$ and $P = \{t_0 < t_1 < \cdots < t_n\}$ is a partition of $[a,b]$ with $\|P\| < \delta$ and $\tau_k\in [t_{k-1},t_k]$ for all $k$, then $P \in \mathscr{P}_m$ and $\sum f(\tau_k) [\gamma(t_k) - \gamma(t_{k-1})] \in F_m\subset B(I, \epsilon)$, which implies $\lvert I - \sum f(\tau_k)[\gamma(t_k) - \gamma(t_{k-1})\rvert < \epsilon$.

Thanks Euge ...

Still reflecting on this ...

Peter

 

[Math Amateur]

Thanks Euge ...

Still reflecting on this ...

Peter