Conditions and classes of examples of variance mixture of normals are given, along with a constructive proof on how to guarantee that a finite variance mixtures of normals is uniformly close (up to a desired tolerance level) to a given infinite variance mixture distribution. We wish to minimize the finite number of terms needed subject to a specified desired tolerance level. The method, which is based on discretizing the mixing measure is presented and illustrated through an example and the infinite and finite mixtures are displayed on the same graph. A new method for estimating the parameters of a variance mixture of normals is also introduced. The new method is based on minimizing the squared distance between the estimated density and the corresponding density computed by discerizing the mixture over a predetermined grid of R values and a grid of X values. This method looks promising especially for modeling data.