I am trying to prove that the ring of $p$-adic integers contains no zero-divisors.
Let $(a_n)$ and $(b_n)$ be $p$-adic integers (so $a_i\in\mathbb{Z}/p^i$ and $a_{n+1}\equiv a_n \pmod{p^n}$ and same for $b$'s).
If these numbers are non-zero, let $a_s$ be the first term in $(a_n)$ which is not $0$ and $b_r$ be the first term in $(b_n)$ which is non-zero.
We can assume that $s\ge r$.
Since $(a_n)(b_n)=0$, so $a_{2s}b_{2s}\equiv 0\pmod{p^{2s}}$.
So $p^s$ should divide $a_{2s}$ or $b_{2s}$.
If $p^s$ divides $a_{2s}$ then it also divides $a_s$, i.e. $a_s=0$, contradiction.
If $p^s$ divides $b_{2s}$ then it also divides $b_s$, hence to $b_r$, i.e. $b_r=0$, a contradiction.
Is this proof correct?
