In Feferman's paper (located here (pg. 2 of the PDF)), he begins to outline various methods used for attacking the problem of classifying the computable functions by means of hierarchies. In particular, he discusses those hierarchies which are formed by a kind of primitive recursive majorizing function, and provides the following definition:

We write $g \ll f$ and say "$f$ majorizes $g$" if we have that $\exists y \forall x [y < x \implies g(x) < f(x)]$.

Feferman then goes on to claim, with this definition in mind, that there is a hierarchy of computable functions which can be formed by extending $\ll$ into the effective transfinite, and bases his claim on following proposition:

For any effectively enumerated class $E$ of computable functions, one can construct a

computablefunction $f$ which majorizes every member $g$ contained in $E$. [italics mine]

My question is the following: Is there a complete proof for this statement in the literature? I imagine that there is a slick 2-line proof which requires little machinery (perhaps simply using diagonalization or a subtle application of the recursion theorem). But since the statement was given without proof, I feel as if I only have cheap, ad-hoc methods for solving related questions, such as how one might go about extending $\ll$, how to construct a computable $f$ for every $g \in E$, etc.

A reference or a quick proof would be greatly appreciated.