We give an elementary proof of the following theorem on definability of Hausdorff limits of one parameter families of definable sets: let A   R×Rⁿ be a bounded definable subset in o-minimal structure on (R,+,) such that for any y∈(0,c), c>0, the fibre Ay x y x A := { Rⁿ : (y , x) } is a Lipschitz cell with constant L independent of y. Then the Hausdorff limit lim y A_y →0 exists and is definable.

We give an elementary proof of the following theorem on definability of Hausdorff limits of one parameter families of definable sets: let A   R×Rⁿ be a bounded definable subset in o-minimal structure on (R,+,) such that for any y∈(0,c), c>0, the fibre Ay x y x A := { Rⁿ : (y , x) } is a Lipschitz cell with constant L independent of y. Then the Hausdorff limit lim y A_y →0 exists and is definable.