Let $U=\mathbb{Z}^{\ast}_p$ be the group of p-adic units. For every $n\geq 1$ put $U_n=1+p^n\mathbb{Z}_p$. This is the kernel of the homomorphism $\varepsilon_n:U\to (\mathbb{Z}/p^n\mathbb{Z})^{\ast}$, $\varepsilon_n(x)=x_n$
I want to show, that
1) $\varepsilon_n$ is indeed a homomorphism and
2) that the kernel is given by $1+p^n\mathbb{Z}_p$
Ad 1)
Let $x,y\in U$, then $\varepsilon(xy)=x_ny_n$ since the multiplication in $\mathbb{Z}_p^\ast$ works "by components". From there we proceed: $x_ny_n=\varepsilon_n(x)\varepsilon_n(y)$ and are done.
So $\varepsilon_n$ is indeed a homomorphism.
Ad 2)
$\ker(\varepsilon_n)=\{x\in U|~x_n\equiv 1\mod p^n\}=1+p^n\mathbb{Z}_p$
Is this really that easy, or do I have to be more precise? Espacially for 2). Thanks in advance.