  Damped Ly$\alpha$ systems provide possibly the most significant evidence for
early structure formation, and thus a stringent constraint on the Cold + Hot
Dark Matter (CHDM) cosmology. Using the numbers of halos in N-body simulations
to normalize Press-Schechter (PS) estimates of the number densities of
protogalaxies as a function of redshift, we find that CHDM with
$\Omega_c/\Omega_\nu/\Omega_b = 0.6/0.3/0.1$ is compatible with the damped
Ly$\alpha$ data only at $z<2.5$, but that it is probably incompatible with data
at $z>3$. The predictions of CHDM are quite sensitive to the neutrino fraction.
We find that $\Omega_c/\Omega_\nu/\Omega_b = 0.725/0.20/0.075$ is compatible
with the $z>3$ data. With one massive neutrino species, this corresponds to
lowering the neutrino mass from 7.0 to 4.7 eV, for $H_0=50\kmsMpc$ and
$T=2.726$ K. By analysing our numerical simulations with different resolutions
and box sizes as well as those of Ma \& Bertchinger (1994), we show that for
the CHDM models with $\Omega_\nu$=0.2--0.3 the PS approximation should be used
with Gaussian filter with $\delta_c=1.3-1.4$ if one tries to recover the total
mass of a collapsed halo and to include nonlinear effects, due to waves both
longer and shorter than those within the simulation box.
