  It has been suggested by Timmermans [Phys. Rev. Lett. {\bf 87}, 240403
(2001)] that loss of fermions in a degenerate system causes strong heating. We
address the fundamental limit imposed by this loss on the temperature that may
be obtained by sympathetic cooling of fermions by bosons. Both a quantum
Boltzmann equation and a quantum Boltzmann \emph{master} equation are used to
study the evolution of the occupation number distribution. It is shown that, in
the thermodynamic limit, the Fermi gas cools to a minimal temperature $k_{{\rm
B}}T/\mu\propto(\gamma_{{\rm loss}}/\gamma_{{\rm coll}})^{0.44}$, where
$\gamma_{{\rm loss}}$ is a constant loss rate, $\gamma_{{\rm coll}}$ is the
bare fermion--boson collision rate not including the reduction due to Fermi
statistics, and $\mu\sim k_{{\rm B}}T_{{\rm F}}$ is the chemical potential. It
is demonstrated that, beyond the thermodynamic limit, the discrete nature of
the momentum spectrum of the system can block cooling. The unusual non-thermal
nature of the number distribution is illustrated from several points of view:
the Fermi surface is distorted, and in the region of zero momentum the number
distribution can descend to values significantly less than unity. Our model
explicitly depends on a constant evaporation rate, the value of which can
strongly affect the minimum temperature.
