The depth of an ideal with a given hilbert function

Let A = K[x₁,⋯,x_n] denote the polynomial ring in n variables over a field K with each deg X₁ = 1. Let I be a homogeneous ideal of A with I ≠ A and H_a/i the Hilbert function of the quotient algebra A/I .Given a numerical function H : ℕ → ℕ satisfying H = H_a/i for some homogeneous ideal I of A,we write A _h for the set of those integers 0 ≤ r ≤ n such that there exists a homogeneous ideal I of A with H_a/i = H and with depth A/I = r. It will be proved that one has either Ah = {0,1,⋯,b} for some 0 ≤ b ≤ n or A|_H | = 1.