I am working through Gouvêa's p-adic Numbers: An Introduction, and Problem 94 is to verify that the following forms a short exact sequence $$ 0\to\mathbb{Z}_p\to\mathbb{Z}_p\to\mathbb{Z}/p^n\mathbb{Z}\to0 $$ where $\mathbb{Z}_p$ is the p-adic integers, $\mathbb{Z}_p=\{x\in \mathbb{Q}_p:|x|\leq 1\}$, and the first map is given by multiplication by $p^n$. I understand why $0\to\mathbb{Z}_p\to\mathbb{Z}_p$ is exact, as it is multiplication in a field. What I am unsure of is what the map $\mathbb{Z}_p\to\mathbb{Z}/p^n\mathbb{Z}$ is.
In the hint, Gouvêa states that this map comes out of the proof of Proposition 3.3.3; unfortunately, there is no such proposition. My only idea is that I think I can write $\mathbb{Z}_p$ as a disjoint union $$ \mathbb{Z}_p=\cup_{k=0}^{p^n-1}B\left(k,\frac{1}{p^n}\right), $$ with $B\left(k,\frac{1}{p^n}\right)$ the open ball of radius $\frac{1}{p^n}$ around $k$, so then I can map $\mathbb{Z}_p\to\mathbb{Z}/p^n\mathbb{Z}$ in the obvious way.
Is this the correct map? If it is, is there a better way of thinking about this? It seems like a pretty roundabout way of defining this, and I don't want to move on in the book without being sure of what the map should be. Thanks for your help.
The term on the RHS is in fact $\mathbb{Z}_p/ p^n \mathbb{Z}_p$. Now you have to show that we also have an isomorphism $$\mathbb{Z}/ p^n \mathbb{Z}\to \mathbb{Z}_p/ p^n \mathbb{Z}_p$$ coming from the inclusion $\mathbb{Z} \subset \mathbb{Z}_p$. The injectivity is easy, the surjectivity comes from the fact that $$\mathbb{Z} + p^n \mathbb{Z}_p = \mathbb{Z}_p$$ ( not direct). This is equivalent to: for every $a \in \mathbb{Z}_p$ there exists $z \in \mathbb{Z}$ so that $d(a,z) \le \frac{1}{p^n}$. (This statement, for all $n$, is equivalent to the density of $\mathbb{Z}$ in $\mathbb{Z}_p$)