I need to show that an infinite recursively enumerable subset of $\Bbb N$ must be the range of a one-to-one recursive function.
A theorem tells me that for a recursively enumerable set, it is either empty or the range of a recursive function.
EDIT:
I think to make it the range of a one-to-one recursive function, what I should do is trying to get a one-to one recursive function from the recursive function we obtained from the theorem I mentioned above. I think maybe I may define
Define function $g$ as:
$g(0)=f(0)$
$g(n+1)=f(min_{x\in\Bbb N} \lbrace f(x)\ne f(n) \forall n<x\rbrace)$
Then $g$ is primitive recursive and hence a recursive function . It is a total function because as $S$ is infinite, we can always find some $x$ such that $f(x)\ne f(n)\forall n<x$ and thus the minimalization is always defined.
Is that correct? Thanks advance for any help!