  In this first paper we begin the application of variational methods to
renormalisable asymptotically free field theories, using the Gross-Neveu model
as a laboratory. This variational method has been shown to lead to a
numerically convergent sequence of approximations for the anharmonic
oscillator. Here we perform a sample calculation in lowest orders, which shows
the superficially disastrous situation of variational calculations in quantum
field theory, and how in the large-$N$ limit all difficulties go away, as a
warm up exercise for the finite-$N$ case and for QCD.
