The existence of Lagrange multipliers at a solution of a nonlinear optimization problem constitutes one of the cornerstones of modern optimization theory, with many important consequences for guiding algorithmic procedures towards a solution, defining stopping criteria, performing stability analysis, and several other aspects. However, the proof of this result is often intricate, relying on non-trivial tools such as Farkas’ Lemma, duality theory, or the implicit function theorem. In this paper, we present a concise and accessible proof of the existence of Lagrange multipliers for nonlinear optimization problems with conic constraints, suitable for advanced undergraduate students or early graduate students. Using only elementary facts about sets and sequences together with Weierstrass’ extreme value theorem, our approach employs a penalization technique combined with basic properties of the projection onto closed convex cones, which are presented in detail. The main result establishes that, under Robinson’s constraint qualification, every local solution admits a Lagrange multiplier satisfying the Karush/Kuhn–Tucker (KKT) conditions.