# Bolzano-Weierstrass Theorem and Pareto Efficient Allocation

Wikipedia says 'The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.' However, I couldn't find a compelling simple proof for this theorem anywhere. Could someone provide a proof for the theorem in a simple fashion that is understandable for Undergrad students in Economics who have done a course in Analysis?

• In order to understand any of the proofs you have to grasp either the concept of "nested intervals" ( en.wikipedia.org/wiki/Nested_intervals ) or the concept of a subsequence. If you get those the proof is straight foreward. Anyway these arent complicated. They are taught in intro to analysis in all undergrad programs in math. – Grada Gukovic Jul 22 at 9:57
• Perhaps insofar as Bolzano-Weierstrass theorem is used to prove the extreme value theorem, which in turn can be invoked to easily establish the existence of Pareto optimality. See also: quora.com/… – Herr K. Jul 22 at 18:08