The reason people mention those is because your line of argument makes absolutely no sense, so that part gets ignored.
Elliptic curves (more generally, Siegel's theorem) is truly the best way to understand the finiteness of integer points. Calculus methods are completely inadequate for the task. If you wrote out your proof formally you would see that. But instead you posted a sketchy version with nothing defined properly.
Discussion between Jim and Erick Wong
Discussion between Jim and Erick Wong