We had a question on this site for Fundamental equations in economics. What I'd like to know is what are the most important proofs needed in economics?
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;