  We establish rigorous upper and lower bounds for the speed of pulled fronts
with a cutoff. We show that the Brunet-Derrida formula corresponds to the
leading order expansion in the cut-off parameter of both the upper and lower
bounds. For sufficiently large cut-off parameter the Brunet-Derrida formula
lies outside the allowed band determined from the bounds. If nonlinearities are
neglected the upper and lower bounds coincide and are the exact linear speed
for all values of the cut-off parameter.
