Recursively enumerable but non recursive subset of an infinte recursive language How can we show that, for every infinite recursive language, it has a subset that is recursively enumerable but not recursive?

I think we need to show there's a list of natural numbers that can't be ordered but I don't know how to build such a subset.

Is there another way of arguing or another type of argument in order to prove such subset exists?
Hint. Let $S$ be any infinite recursive language. By considering, e.g., the lexicographic order, we can talk of the first, second, third, etc. strings in&nbsp;$S$. Use this indexing to define what you need to define.