The depth of an ideal with a given Hilbert function

Let $A = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\operatorname{deg} x_i = 1$. Let $I$ be a homogeneous ideal of $A$ with $I \neq A$ and $H_{A/I}$ the Hilbert function of the quotient algebra $A / I$. Given a numerical function $H : {\mathbb{N}} \to {\mathbb{N}}$ satisfying $H=H_{A/I}$ for some homogeneous ideal $I$ of $A$, we write $\mathcal{A} _H$ for the set of those integers $0 \leq r \leq n$ such that there exists a homogeneous ideal $I$ of $A$ with $H_{A/I} = H$ and with $\operatorname{depth} A / I = r$. It will be proved that one has either $\mathcal{A}_H = \{ 0, 1, \ldots, b \}$ for some $0 \leq b \leq n$ or $\vert{\mathcal{A}}_H\vert = 1$.