A new proof of the Lyapunov convexity theorem

Because of its importance, many proofs of the Lyapunov Theorem have been given. Nevertheless, the proofs appearing in the literature are rather involved or employ sophisticated theorems from functional analysis, such as the Krein-Milman Theorem, or compactness theorems in infinite-dimensional spaces. A new proof of the Lyapunov Theorem is given, based on the Shapley-Folkman Theorem, that does not require any tools of functional analysis.