A set $X\subseteq \mathbb{N}$ is $n$-c.e. if there is a total recursive guessing procedure $g(x,s)$ so that $$ g(x,0) = 0,\ \lim_s g(x,s) = X(x) $$ and the number of times $g$ changes its mind on a given $x$ i.e. $|\{s\ | \ g(x,s)\neq g(x,s+1)\}|$ is $\leq n$.
We want to show that there is an $n+1$-c.e. set which is not $n$-c.e.
I have a hint which begins "Fix $n$. First specify a method to effectively list all $n$-c.e. sets".
Given this hint I can diagonalise in the (relatively) obvious way, but I don't see a way to effectively list all such (indices for) sets. Am I supposed to use the fact that an $n$-c.e. set can be expressed as a difference of unions of difference etc of c.e. sets?
Well it seems I do precisely what I suggested. Every $n$-c.e. set is of the form $(W_{e_1}\setminus W_{e_2})\cup(W_{e_3}\setminus W_{e_4})\cdots$ and so we can code this set as the tuple $(e_1,\ldots,e_n)$. We can turn such a code into a recursive guessing procedure for the set, which changes its mind at most $n$ times, uniformly in $(e_1,\ldots,e_n)$ then I can diagonalise.