A Formal proof of Green Theorem

Question

I want to go through the formal proof of Green theorem on a regular, simple and closed curve oriented counterclockwise and the vector space $F$ is a continuously differentiable vector field such that $F: R^2->R^2$ defined on some open set containing C. I want to proof it by refinement in small rectangles. I proved that the cancelation does work, that means that if I have two adjacent rectangles I don't need to care about the edge in the middle. But I have trouble to use $\epsilon$ and $\delta$ to prove that the error is negligible. That means I want to proof that the for the compact set C, there is a compact set C' contained in C, such that C' is at most $\epsilon $ distance away from C. And my refinement can cover C' but contained in C. Could any one give me some hint on it? I cannot find any information on how to deal with the refinement in the proof of Green theorem but only general idea. I really want to go through the detail about it. Thank you so much for your help