I've read somewhere that all clopen subsets of a profinite group $$G \simeq \varprojlim\left(G_i, f_{ij}:G_i \to G_j\right)_{i,j \in I}$$ are exactly the preimages of subsets of the $G_i$'s. It's easy to see that subsets of $G$ of the latter form are clopen. But I'm not so sure about the other direction.
So my question is: is the above description of the clopen sets of $G$ true? If so, how? And if not, what's the best we can do?
Yes, this is true. Let me call a subset of $G$ good if it is the preimage of a subset of $G_i$ for some $i$. First, note that the good sets are closed under finite unions and intersections: if $U$ the preimage of a subset of $G_i$ and $V$ is the preimage of a subset of $G_j$, then $U\cap V$ and $U\cup V$ are preimages of subsets of any $G_k$ that maps to both $G_i$ and $G_j$ in the inverse system. Now by definition, the good sets form a subbasis for the topology on $G$. Since they are closed under finite intersections, actually the good sets form a basis. Thus any clopen set is a union of good sets. But a clopen set is compact, so it is actually a finite union of good sets, and hence is itself good.