How to show that any path $\gamma:[a, b]\rightarrow\mathbb C$ is rectifiable and that $L(\gamma)=\int_{a}^{b}|\gamma'(t)|dt$. How to show that any path $\gamma:[a, b]\rightarrow\mathbb C$ is rectifiable and that $L(\gamma)=\int_{a}^{b}|\gamma'(t)|dt$.
Definition:
Consider a partition $P,a=t_0\lt t_1\lt \ldots\lt t_n=b\; of\; [a, b]$ and define
$L(P)=\sum_{i=0}^{n-1}|\gamma(t_{i+1}-\gamma(t_i)|$
The length of the curve , $L(\gamma)= sup\{L(P):P \;is\; a\; partition \;of\;[a,\;b]\}$
If $L(\gamma)\lt\infty$ then $\gamma$ is said to be rectifiable.

Use triangle inequality $|\sum_{i=0}^{n-1}(\gamma(t_{i+1})-\gamma(t_i))|\leq\sum_{i=0}^{n-1}|\gamma(t_{i+1})-\gamma(t_i)|=L(\gamma)$.
$|\sum_{i=0}^{n-1}(\gamma(t_{i+1})-\gamma(t_i))|=|\int_{t_0}^{t_n}\gamma'(t') dt'|, $
$|\gamma(t_{i+1})-\gamma(t_i)|=|\int_{t_i}^{t_{i+1}}\gamma'(t') dt'| \leq \int_{t_i}^{t_{i+1}}|\gamma'(t')| dt'$.
You get the final inequality
$|\sum_{i=0}^{n-1}(\gamma(t_{i+1})-\gamma(t_i))|\leq L(\gamma) \leq \sum_{i=0}^{n-1} \int_{t_i}^{t_{i+1}}|\gamma'(t')| dt'$.
Use the mean value Theorem for the Limit and you are done.
