I am having some trouble showing the following: If P is productive, then P contains an infinite r.e. set W. (It is part of a theorem from Soare's "Recursively Enumerable Sets and Degrees).
The proof in Soare goes as follows:
Let $W_n=\emptyset$, and $W_{h(x)}=W_x\cup \{p(x)\}$. Define \begin{align*} W=\{p(n), ph(n),ph^2(n),...\}. \end{align*}
And that is it.
Note that h is a recursive (total) function, that p is an injective, total, recursive, productive function for the set P.
I understand why $W\subseteq P$, and that $W$ is an inifite set. I can not see clearly why $W$ is r.e. (recursively enumerable). Can anyone help me?