In Seer's "A Course in Arithmetic" he states:
Let $U = \Bbb Z_p^*$ be the group of $p$-adic units. We define a homomorphism $\epsilon_n : U \rightarrow (\Bbb Z/p^n \Bbb Z)^*$. Then the $U_n = 1 + p^n \Bbb Z_p$, and this is the kernel of the homomorphism.
My question is why is this the kernel?
I have tried visualizing $\epsilon_n$ as a truncation but I still can't get it.
As a sideline question could someone please confirm for me that EVERY $p$-adic integer is also a unit? I couldn't find a clear answer online.