For a set $A \subseteq \omega$, let $[A]^n$ denote the set of subsets of $A$ of size $n$. I am trying to prove Ramsey's Theorem, and it seems like the following fact is used in the proof I am reading (Jech 9.1):
Suppose for every function $F:[\omega]^n \to \{ 1, \ldots, k \}$, there exists an infinite set $H \subseteq \omega$ such that F is constant on $[H]^n$. Then for any infinite subset $A\subseteq \omega$, and any function $F_A: [A]^n \to \{1, \ldots, k\}$, there exists an infinite subset $H_A \subseteq A$ such that $F_A$ is constant on $H_A$.
Is this true, and how do I prove it? I can't work out how to define $H_A$. If I extend $F_A$ to all of $[\omega]^n$ somehow, I know I can deduce there is an infinite set $H$ such that the extension is constant on $[H]^n$, but I don't know how to reduce this to an infinite set $H_A \subseteq A$, or how to extend the function. If I just define $H_A := A \cap H$, then I don't see why this is necessarily infinite.
Any help would be appreciated.