  Previous large $N$ calculations are combined with numerical work at $N=4$ to
show that the Minimal Standard Model will describe physics to an accuracy of a
few percent up to energies of the order 2 to 4 times the Higgs mass, $M_H$,
only if $M_H \le 710\pm60 ~ GeV$. This bound is the result of a systematic
search in the space of dimension six operators and is expected to hold in the
{\it continuum}. Given that studying the scalar sector in isolation is already
an approximation, we believe that our result is sufficiently accurate and that
further refinements would be of progressively diminishing interest to particle
physics. Note: The full ps file of this preprint is also available via
anonymous ftp to ftp.scri.fsu.edu. To get the ps file, ftp to this address and
use for username "anonymous" and for password your name. The file is in the
directory pub/vranas (to go to that directory type: cd pub/vranas) and is
called num_hig.ps (to get it type: get num_hig.ps)
