  The Eisenbud-Green-Harris conjecture states that a homogeneous ideal in
k[x_1,...,x_n] containing a homogeneous regular sequence f_1,...,f_n with
deg(f_i)=a_i has the same Hilbert function as an ideal containing x_i^{a_i} for
1 \leq i \leq n. In this paper we prove the Eisenbud-Green-Harris conjecture
when a_j> sum_{i=1}^{j-1} (a_i-1) for all j>1. This result was independently
obtained by the two authors.
