  Generic $U(1)^2$ 4-d black holes with unbroken $N=1$ supersymmetry are shown
to tend to a Robinson-Bertotti type geometry with a linear dilaton and doubling
of unbroken supersymmetries near the horizon. Purely magnetic dilatonic black
holes, which have unbroken $N=2$ supersymmetry, behave near the horizon as a
2-d linear dilaton vacuum $\otimes \, S^2$. This geometry is invariant under 8
supersymmetries, i.e. half of the original $N=4$ supersymmetries are unbroken.
The supersymmetric positivity bound, which requires the mass of the 4-d dilaton
black holes to be greater than or equal to the central charge, corresponds to
positivity of mass for a class of stringy 2-d black holes.
