Existence of Pareto Optima

Lemma. Assumptions 1 and 3 imply $A$ is closed.

Proof. Take a sequence $(x^k_n,y^k_m)$ of feasible production plans with limit $(x,y)$. Each $x^k_n$ and $y^k_n$ are in their respective consumption and production sets $X_n$ and $Y_m$. These sets are closed, and therefore contain their limits. ∎