Question about Prop 7.31 in Milne's Algebraic Number Theory
Solution 1:
By construction $a_{n+1} \equiv a_n \pmod{\pi^{n+1}}$. In the $\pi$-adic topology this means that $a_{n+1}$ and $a_n$ get closer and closer together as $n$ increases. In other words, the sequence $(a_n)$ is Cauchy. Finally, remember that the space $A$ is complete, hence every Cauchy sequence is convergent. The uniqueness of the limit follows as the space is also Hausdorff.