A geometric proof of the definability of Hausdorff limits

Abstract.We give a geometric proof of the following well-established theorem for o-minimal expansions of the real field: the Hausdorff limits of a compact, definable family of sets are definable. While previous proofs of this fact relied on the model-theoretic compactness theorem, our proof explicitly describes the family of all Hausdorff limits in terms of the original family.