  In this note we show that a set is provably $\Delta^0_2$ in the fragment
$I\Sigma_n$ of arithmetic iff it is $I\Sigma_n$-provably in the class
$D_\alpha$ of $\alpha$-r.e. sets in the Ershov hierarchy for an $\alpha
<_{\epsilon_0} \omega_{1+n}$, where $<_{\epsilon_0}$ denotes a standard
$\epsilon_0$-ordering. In the Appendix it is shown that a limit existence rule
$(LimR)$ due to Beklemishev and Visser becomes stronger when the number of
nested applications of the inference rule grows.
