The Welfare Theorems

Lemma. If $\succeq_n$ is locally non-satiated at bundle $x^*_n$ which is preference-maximal on the set $\{x_n\in X_n:px_n\leq px^*_n\}$, and if $x'_n\succeq_n x^*_n$, then $px'_n\geq px^*_n$.

Proof. Since $\succeq_n$ is locally non-satiated at $x'_n$, there is a sequence of consumption bundles $x^k_n$ with limit $x'_n$ such that $x^k_n\succ_nx'_n$. Transitivity implies that $x^k_n\succ_nx^*_n$. Preference maximality implies that $px^k_n>px^*_n$. Taking limits, $px'_n\geq px^*_n$. ∎