To provide another answer (or rather a hint, as it is an exercise): You can indeed, as Richard suggested, use the characterization theorem using multicomplexes. More elementary, you can use the following idea: the contribution to h_i appears when along a shelling, the minimal new face is of cardinality face. Consider the complementary face, of cardinality d-i (if the complex is of dimension d-1). This is the minimal "old" face. 

Now, there are at least two contributions to h_i. You can assume that the minimal old faces are disjoint (by passing to the link of a intersection). Similarly, you can pass to links and assume that the minimal new face is disjoint. You can now apply your argument to each face separately.