Solution
= Solution
Suppose $G$ is topologically generated by $d$ elements. An open subgroup $H$ of index $n$ gives a continuous transitive <coset action>
$$
G\longrightarrow S_n.
$$
A continuous homomorphism $G\to S_n$ is determined by the images of the $d$ topological generators, so there are at most $|S_n|^d$ such homomorphisms. Each has only finitely many point stabilizers. Therefore $G$ has only finitely many open subgroups of index $n$.