Easy Proofs of L\"owenheim-Skolem Theorems by Means of Evaluation Games

We propose a proof of the downward L"owenheim-Skolem that relies on strategies deriving from evaluation games instead of the Skolem normal forms. This proof is simpler, and easily understood by the students, although it requires, when defining the semantics of first-order logic to introduce first a few notions inherited from game theory such as the one of an evaluation game.