Let $A\subset B$ be an extension of discrete valuation rings and let $p$ and $P$ be the non-zero prime ideals of $A$ and $B$ respectively.
So I can write $pB=P^m$ for some $m>0$.
I form the inverse limit $\varprojlim B/p^nB$ with respect to the inverse system $\{B/p^nB,\pi_m^n\}$ where $\pi_m^n$ is the residue class map $B/p^nB\to B/p^mB$ for $n\geq m$.
Then is it true that $\varprojlim B/p^nB=\varprojlim B/P^n$?