Is it true that if $G = \varprojlim G_i$ is a profinite group such that every $G_i$ has a generator set $S_i$ whose cardinality is uniformly bounded for all $i$, then $G$ is (topologically) finitely generated?
I found it in the survey linked here: https://arxiv.org/pdf/math/0703885.pdf
forth line, page number 3, with no reference and I could not prove it nor disprove it.
The converse is of course true and this sounds intuitive for me, from the perspective that $G$ tends to be the limit of the $G_i$'s. However, a clear obstruction is that no every lift of an element in $G_i$ should work to topologically generate everything.
Let $\phi_{ji}:G_j \longrightarrow G_i$ and $\phi_{i}:G\longrightarrow G_i$ are natural maps related to a given inverse limit system. Here all $G_i$'s are finite groups. It seems you need to assume the minimal number of generators of $\phi_{i}(G)$ (for all $i$) is always smaller than or equal to those of $G_i$'s (I don't know whether this condition always holds.). Then proceed as follows:
Step1: We may assume that all inverse limit maps $\phi_{ji}:G_j \longrightarrow G_i$ are surjective. say $n$ the uniform minimal bound of the number of minimal generators of $G_i$.
Step2: Define $\Psi_i:=\{S\subset G_i : <S>=G_i\}$ (note $\Psi_i$ is a subset of the power set of $G_i$.). Here $<S>=G_i$ means $S$ generates $G_i$. Then $\Psi_i$ is finite for all $i$ and $\{\Psi_i\}$ forms an inverse system where the inverse limit maps are naturally induced by $\phi_{ji}$'s. Then $\varprojlim_i \Psi_i$ is a nonempty set by the nonemptyness of profinite space (which is induced from the fact that a product space of compact spaces is again compact). Choose $(S_i)\in \varprojlim_i \Psi_i$. Since $|S_i|$ (cardinality of $S_i$) is equal to $n$ and $\phi_{ji}:S_j \longrightarrow S_i$ is surjective, $\phi_{ji}:S_j \longrightarrow S_i$ is a bijection for sufficiently large $i$. It follows that $(S_i)$ is a finite set with cardinality $n$. Note that $(S_i)$ is a topological generator set of $G$. Done.
I recommend the book `Profinite groups' by Ribes & Zalesskii for various properties of profinite groups.