Existence of Pareto Optima

Step 2: The theorem is proved by showing that the existence of a sequence with an unbounded production plan violates the irreversibility of production, Assumption 3. With no loss of generality, by passing to subsequences, we can assume that $y^k_1$ is an unbounded sequence of production plans for firm 1, that $||y^k_1||=\max||y^k_m||$ for all $m$ and $k$. For all $m$ and $k$ define $\bar y^k_m= ||y^k_1||^{-1}y^k_m$. Each $\bar y^k_m$ is a convex combination of $y^k_m$ and 0, and so in $Y_m$. Furthermore, each $||\bar y^k_m|| \leq ||\bar y^k_1||=1$. Again by passing to a subsequence if necessary, we can assume that the sequences $\bar y^k_m$ each converge to a limit $\bar y_m$. Since the sets $Y_m$ are closed, each $\bar y_m\in Y_m$.

Feasibility of $(x^k,y^k)$ implies that \begin{equation*} \omega+\sum_my^k_m=\sum_mx^k_m\leq\sum_n x'_n \end{equation*}