  We show that in dilute boson-fermion mixtures with fermions in two internal
states, even when the bare fermion-fermion interaction is repulsive, the
exchange of density fluctuations of the Bose condensate may lead to an
effective fermion-fermion attraction, and thus to a Cooper instability in the
s-wave channel. We give an analytical method to derive the associated $T_c$ in
the limit where the phonon branch of the Bogoliubov excitation spectrum of the
bosons is important. We find a $T_c$ of the same order as for a pure Fermi gas
with bare attraction.
