Short exact sequence with p-adic integers and $\mathbb{Z}/p^n\mathbb{Z}$ 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.
 A: 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 $\mathbb{Z}_p$) 
A: The explicit map I think is from the previous proposition as the author mentioned. In my edition (Second edition), Proposition 3.3.4 says the following:
Given $x \in \mathbb{Z}_p$ and $n \geq 1$, there exists $\alpha \in \mathbb{Z}, 0 \leq \alpha \leq p^n-1$ such that $|x-\alpha|\leq p^{-n}$. The integer $\alpha$ with these properties is unique. 
This is the density mentioned by another user with an extra condition on the integer.
Consider the map $x \mapsto [\alpha]$ where $[\alpha]$ is the equivalence class of $\alpha$ in $\mathbb{Z}/p^n\mathbb{Z}$
If $x,y \in \mathbb{Z}_p$ and $\alpha, \beta$ are the corresponding integers mentioned in the Proposition. To show that the above map is indeed a homomorphism, first see that
If $\alpha \equiv \beta ~(\bmod{~p^n}~)$ and $|x-\alpha| \leq p^{-n}$, then
$$ |x-\beta| = |x-\alpha+p^nm| \leq \max(|x-\alpha|, |p^nm|) \leq p^{-n} $$
Here we used that $|p^nm| = |p^n||m| \leq |p^n| = p^{-n}$.
$$ |x+y -(\alpha+\beta)| \leq \max (|x-\alpha|,|y-\beta|) \leq p^{-n} $$
Hence if we choose $\gamma \equiv \alpha+\beta~(\bmod{~p^n}~)$ with the property that $0 \leq \gamma \leq p^n-1$, we obtain
$$ |x+y-\gamma| \leq p^{-n} $$
By uniqueness, we get $[\alpha] + [\beta] = [\gamma]$.
Multiplication works in a similar way. Surjectivity is clear as $\mathbb{Z} \subset \mathbb{Z}_p$.
