Two dimensional non-linear sigma models as a limit of the linear sigma models

We show how to obtain the O(N) non-linear sigma model in two dimensions as a strong coupling limit of the corresponding linear sigma model. In taking the strong coupling limit, the squared mass parameter must be given a specific coupling dependence that assures the finiteness of the physical mass scale. The relation discussed in this paper, which applies to the renormalized theories as opposed to the regularized theories, is an example of a general relation between the linear and non-linear models in two and three dimensions.