This is a question related to this.
Let $G= \mathbb Z / p \mathbb Z$ for some prime $p$. Let $A = \bigoplus_{n\in \mathbb N} G$, that is, all sequences in $G$ with all but finitely many terms zero. Now I'm interested in the $p$-adic completion of this group. The answer is supposed to be that the completion of $A$ is $A$ itself. But I don't see how that is true. The completion is the same as the inverse limit, which is a subset of the product $\prod_{n \in \mathbb N} A$. So it's sequences of sequences. But consider the sequence $s_n$ where the first $n$ terms are one. Then clearly this is a Cauchy sequence in the $p$-adic norm but its limit is not in $A$. So how can $A$ be complete? Thanks for your help.
Edit
$|x|_p = \frac{1}{\text{highest power of } p \text{ that divides } x}$, $|0|_p = 0$