In the proof of the Green theorem (in the last step: any region can be approximated as closely as we want by a sum of rectangles), I need to prove the following result:
Let $\gamma$ be a Jordan curve (closed and simple) on $\mathbb{R}^2$. Suppose that $\gamma$ is parametrized by $\gamma: [a, b]\to \mathbb{R}^2$ and is piecewise regular (this means that, It can be written as a concatenation of $C^1$ paths with not null speed in all its points). Prove that there exist $\delta>0$ such that, if $a = t_0 < t_1 < \cdots < t_N = b$ is a partition of $[a, b]$ with $$t_i − t_{i−1}<\delta,\quad for \ all \quad i = 1,\cdots,N$$ then the polygonal curve $P$ wich joints the points $\gamma (t_1), \gamma(t_2),\cdots,\gamma(t_N) = \gamma(t_0)$ is a Jordan curve too.