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We show how to perform systematically improvable variational calculations inthe $O(2N)$ Gross-Neveu model for generic $N$, in such a way that allinfinities usually plaguing such calculations are accounted for in a waycompatible with the perturbative renormalization group . The final point is ageneral framework for the calculation of non-perturbative quantities likecondensates, masses etc$\ldots$, in an asymptotically free field theory. Forthe Gross-Neveu model, the numerical results obtained from a ``2-loop'' down tolow values of $N$.Comment: 35 pages, Latex, 5 figures. Correction of a trivial mistake in the  Pad\'e approximant numerical program leads to a significant improvement of  the variational results for the mass gap, for arbitrary N. A more complete  comparison of different Pad\'e approximants is performed. Text modified  according to these new results, one figure and two references adde

Variational Solution of the Gross–Neveu Model: Finite N and Renormalization