That looks about right. You just have to make sure that the intermediate points you choose for the triangle inequality all lie within the domain U, where f is defined and that might also be the reason why U is required to be an open set, because then it is ensured you can always find such intermediate points, that lie within U.

But you also need the property that the partial derivatives are bounded. From this you can conclude that the function is locally lipschitz continous along the directions of the axis and you have to use this somehow in your inequality.