  Extreme black holes have been argued to be unstable, in the sense that under
linearized gravitational perturbations of the extreme Kerr spacetime the Weyl
scalar $\psi_4$ blows up along their event horizons at very late advanced
times. We show numerically, by solving the Teukolsky equation in 2+1D, that all
algebraically-independent curvature scalar polynomials approach limits that
exist when advanced time along the event horizon approaches infinity.
Therefore, the horizons of extreme black holes are stable against linearized
gravitational perturbations. We argue that the divergence of $\psi_4$ is a
consequence of the choice of a fixed tetrad, and that in a suitable dynamical
tetrad all Weyl scalars, including $\psi_4$, approach their background extreme
Kerr values. We make similar conclusions also for the case of scalar field
perturbations of extreme Kerr.
