The last part of the paper *Located Sets and Reverse Mathematics* [Journal of Symbolic Logic 65 (1999), 1451–1480] by Giusto and Simpson involves a proof as follows:

Given $A$ an effectively immune set, i.e. there exists a recursive function $p$ such that $A$ is infinite and $W_e\subseteq A$ implies $|W_e|< p(e)$, construct a r.e. set as follows:

$$W_{g(e)}=\begin{cases}\text{the first } p(\varphi_e(e)) \text{ elements from A} & \text{if }\varphi_e(e)\downarrow \\\\ \emptyset &\text{otherwise} \end{cases}$$

It was claimed that $g\leq_TA$, but an issue here is should the set be r.e relative to A, namely the set generated should be $W_{g(e)}^A$, which then should give a different index? The definition here seems to require unbounded amount of information about A to be known ahead of time so A should be deemed as an oracle, shouldn't it? Thanks! (BTW, the fact that the resulting set is r.e. not with respect to any nontrivial oracle is crucial in the proof that follows).