I have a few questions about the proof of the following theorem (outlined in red).
Lemma Cauchy-Goursat
Let $\triangle=\triangle(a,b,c)$ be a triangle in an open set $\Omega \subseteq \mathbb{C},\ p\in \Omega,\ f:\Omega\rightarrow \mathbb{C}$ continuous and $f$ analytical on $\Omega \setminus \{p\}$. Then:
$\int\limits_{\partial\triangle}f(z)dz=0$ , where $\partial\triangle$ means on and inside the triangle $\triangle$.
Cauchy Theorem for convex sets
Let $\Omega\subseteq \mathbb{C}$ open and convex, $ p\in\Omega,\ f$ : $\Omega\rightarrow \mathbb{C}$ continuous, $f\in H(\Omega\backslash \{p\})$ ($f$ analytic on $\Omega \backslash \{p\}$. Then there is a $F\in H(\Omega)$ (F analytic on $\Omega$) with $F'(z)=f(z)$ for all $ z\in\Omega$.
It also folows that $\displaystyle \int\limits_{\gamma}f(z)\mathrm{d}z=0$ for every piecewise continuously differentiable and closed path $\gamma\in\Omega$. (e)
Proof
Fix $ a\in\Omega$ . Because of the convexity of $\Omega$, we have
$$
F(z):=\int\limits_{[a,z]}f(\xi)\mathrm{d}\xi\ (z\in\Omega).\ (a)
$$
$\color{red}{\text{(1) Why (a) follow from }\Omega \text{ being convex ? }}$
Also, because $\Omega$ is convex, we have every triangle $\triangle(a,\ z_{0},\ z)$ in $\Omega\ (z,\ z_{0}\in\Omega). \ \ \ (b)$
$\color{red}{\text{(2) Why is (b) true ?}}$
From the Cauchy-Goursat lemma presented above, we have:
$$
F(z)-F(z_{0})=\int\limits_{[a,z]}f(\xi)\mathrm{d}\xi+\int_{[z_{0},a]}f(\xi)\mathrm{d}\xi=\int_{[z_{0},z]}f(\xi)\mathrm{d}\xi. \ (c)
$$
$\color{red}{\text{(3) How does (c) follow from the Cauchy-Goursat lemma?}}$
For a fixed $ z_{0}\in\Omega$ it follows for $z\in\Omega\backslash \{z_{0}\}$:
$$
\frac{F(z)-F(z_{0})}{z-z_{0}}-f(z_{0})=\frac{1}{z-z_{0}}\int\limits_{[z_{0},z]}(f(\xi)-f(z_{0}))\mathrm{d}\xi \ \ (d)
$$
$\color{red}{\text{(4) Why is in (d): }\displaystyle f(z_0)=\frac{1}{z-z_{0}}\int\limits_{[z_{0},z]}f(z_{0})\mathrm{d}\xi}$
Let $\varepsilon>0$. Because $f$ is in $z_{0}$ continuous, we have a $\delta>0$ with $|f(\xi)-f(z_{0})|<\varepsilon$, if $|\xi-z_{0}|<\delta$. For $ 0<|z-z_{0}|<\delta$ we have:
$$
\left|\frac{F(z)-F(z_{0})}{z-z_{0}}-f(z_{0})\right|<\frac{1}{|z-z_{0}|}L([z_{0},z])\varepsilon=\varepsilon. \ (d)
$$
$L[z_0,z]$ is the length from $z_0$ to $z$.
$\color{red}{\text{(5) In (d): This is the ML-inequality, right? Where does the }\frac{1}{|z-z_{0}|} \text{come from?}}$
So $F'(z_{0})=f(z_{0})$. Because $ z_{0}\in\Omega$ was chosen arbitrarily, we have $F$ analytic in $\Omega$.
$\color{red}{\text{(6) But we still have not shown }\\ \displaystyle \int\limits_{\gamma}f(z)\mathrm{d}z=0.
\text{ This is appears in the statement of the theorem, the last paragraph (see (e) ). How does this follow?}}$
As you can see, I am utterly lost here.
Best Answer
Here is a start.