What are the fundamental economic proofs
I would say the following four:


Nash Equilibrium via Fixed Point (either Kakutani, Brouwer or Tarski, depending on the problem);
Welfare theorems via Hanh-Banach theory (mainly the separating hyperplane theorem);
Utility and Profit Optimization via Weierstrass Theorem, Kuhn-Tucker conditions, and existence of a Lagrangian;
Minimization of Least Squares;