The Welfare Theorems

Define the set $G=\sum_{n\neq k}R(x^*_n)+P(x^*_k)-Y$.This set is convex and $\omega$ is not in $G$ because the allocation is Pareto optimal. Thus there is a vector $p^*$ such that $p^*\omega\leq p^*g$ for all $g\in G$. Since consumer $k$ is locally non-satiated, there is a sequence of consumption plans $x^i_k$ with limit $x^*_k$, each element of which is better for $k$ than $x^*_k$. Then for all $n$ the vector \begin{equation*} g^i=\sum_{n\neq k}x^*_n+x^i_k-\sum_my^*_m \end{equation*} is in $G$, and the sequence $g^i$ converges to \begin{equation*} \omega=\sum_{n\neq k}x^*_n+x^*_k-\sum_my^*_m. \end{equation*} Thus $\omega\in\partial G$ and $\inf\{p^*g:g\in G\}=p^*\omega$.