How do you prove that $\Bbb{Z}_p$ is an integral domain? Let $\Bbb{Z}_p$ be the $p$-adic integers given by formal series $\sum_{i\geq 0} a_i p^i$.  I'm having trouble proving that it's an integral domain.

As $\Bbb Z_p/p^n\Bbb Z_p\cong\Bbb Z/p^n\Bbb Z$, to show $a,b\ne0\Rightarrow ab\ne0$ it suffices to show that $ab\not\equiv0$ mod $p^n$ for a high enough power $p^n$ given $a,b\ne0$ (see vadim's hint in the comments).
