Let $p$ be a prime and let $\{A_n\}_{n > 0}$ be an inverse limit of abelian groups such that $A_n$ is $p^n$-torsion with $A_n/p^{n - 1} \cong A_{n - 1}$ (these isomorphisms are part of the data). Let $A = \varprojlim A_n$ be the inverse limit, which surjects onto each $A_n$. Is it necessarily true that $A/p^n \cong A_n$?
This should be elementary but I can neither prove it nor find a likely to exist counterexample. It is equivalent to ask whether $lim^1$ vanishes for the system $\{ A_n[p^m]\}_{n > 0}$ for each fixed $m$.
