A short simple proof of closedness of convex cones and Farkas' lemma

TL;DR

This paper presents a concise and straightforward proof demonstrating the closedness of finitely generated convex cones, simplifying the geometric proof of Farkas' lemma and deriving it from this result.

Contribution

It introduces a simple proof of the closedness of finitely generated convex cones and derives Farkas' lemma from this, streamlining existing geometric proofs.

Findings

Short proof of convex cone closedness

Simplified derivation of Farkas' lemma

Clarification of geometric proof techniques

Abstract

Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas' lemma. We provide a short simple proof of this fact and (for completeness) derive Farkas' lemma from it using well-known arguments.