So I’m reading a proof a proof for Cauchy’s theorem for triangles. That is if $f:U\rightarrow \mathbb{C} $ is analytic and $\Delta $ is a triangle in open $U $ with boundary $\gamma $ ($U$ contains $\gamma $ too). Then $$\int_{\gamma }f(z)\ dz=0 .$$
The proof basically iteratively splits the triangle into 4 smaller triangles and uses the triangles with the greatest value of the integral. Anyway, you have a point $z_*$ which is in $\overline{\Delta }$ and because $f$ is differentiable there we can say for every $\epsilon >0$ there is a $\delta >0 $ such that if $|z-z_* |<\delta $ then $$f(z)=f(z_*)+f’(z_*)(z-z_*)+R(z,z_*) $$ where $|R(z,z_*)|<\epsilon |z-z_*|.$
So $$|\int_{\gamma^{(n)}}f(z)\ dz |=|\int_{\gamma^{(n)}} R(z,z_*) \ dz| .$$
So the question I’m ask asking is, what happened to $\int f(z_*)\ dz $ and $\int f’(z_*)(z-z_*)\ dz $ ? What’s the justification for them integrals being 0?