Symbolic powers of ideals of generic points in P3

Harbourne and Huneke conjectured that for any ideal I of fat points in PN, its Nr-th symbolic power I(Nr) should be contained in M(N−1)rIr, where M denotes the homogeneous maximal ideal in the ring of coordinates of PN. We show that this conjecture holds for the ideal of any number of simple (not fat) points in general position in P3 and for up to N+1 simple points in general position in PN. As a corollary, we give a positive answer to the Chudnovsky conjecture in the case of generic points in P3.