Existence of Pareto Optima

Lemma. $A$ is bounded.

Proof. Step 1: If a feasible sequence $(x^k,y^k)$ is unbounded, then there is a firm $m$ for whom $y^k_m$ is unbounded. To see this we show that if the production plans are bounded, then the consumption plans are bounded as well. So suppose $(x^k,y^k)$ is a feasible sequence with bounded production. Then $\sum_my^k_m$ is bounded above by some vector $y$. Each $X_n$ is bounded below by a consumption bundle $x'_n$ Each $x^k_p$ is bounded thus: \begin{align*} x'_p\leq x^k_p&\leq\sum_n\omega_n+\sum_my^k_m-\sum_{n\neq p}x^k_p\\ &\leq\sum_n\omega_n+y-\sum_{n\neq p} x'_n \end{align*}