Estimation Lemma for Contour Integrals

Theorem
Let $C$ be a contour.

Let $f: \operatorname{Im} \left({C}\right) \to \C$ be a continuous complex function, where $\operatorname{Im} \left({C}\right)$ denotes the image of $C$.

Then:


 * $\displaystyle \left\vert{ \int_C f \left({z}\right) \ \mathrm dz }\right\vert \le \max_{z \mathop \in \operatorname{Im} \left({C}\right) } \left\vert{f \left({z}\right) }\right\vert L \left({C}\right)$

where $L \left({C}\right)$ denotes the length of $C$.

Proof
By definition of contour, $C$ is a concatenation of a finite sequence $C_1, \ldots, C_n$ of directed smooth curves.

Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

Then: