Existence of Pareto Optima

Existence follows from the serial dictator argument one it shown that the feasible set is non-empty and compact. The feasible set is \begin{equation*} A=\left\{(x,y):x_n\in X_n,y_m\in Y_m,\sum_nx_n\leq\sum_n\omega_n+\sum_ny_m\right\}. \end{equation*} Assumptions 1 and 3 imply that $x_n=\omega_n$, $y_m=0$ for all $n$ and $m$ describes a feasible allocation, so $A$ is non-empty.