  Let K be an abstract elementary class of models. Assume that there are less
than the maximal number of models in K_{\lambda^{+n}} (namely models in K of
power \lambda^{+n}) for all n. We provide conditions on K_\lambda, that imply
the existence of a model in K_{\lambda^{+n}} for all n. We do this by providing
sufficiently strong conditions on K_\lambda, that they are inherited by a
properly chosen subclass of K_{\lambda^+}.
