Sufficiency of McMullen's conditions for $f$-vectors of simplicial polytopes

For convex d-polytope P let ft{P) equal the number of faces of P of dimension i, 0 < i < d 1. f(P) = (f0(P)9 . . . , fd^QP)) is called the f vector of P An important combinatorial problem is the characterization of the class of all /-vectors of polytopes, and in particular of simplicial polytopes (i.e. those for which each facet is a simplex). McMuUen in [5] conjectures a set of necessary and sufficient conditions for (/0, . • . 9fd~i) to be the/-vector of a simplicial d-polytope and proves this conjecture in the case of polytopes with few vertices. We sketch here a proof of the sufficiency of these conditions, and derive in a related way a general solution to an upper bound problem posed by Klee. The /-vectors of simplicial <i-polytopes satisfy the Dehn-Sommerville equations