Let $\{G_i, \phi_{i,j}, i, j \in I\}$ (with a partially ordered $I$) be an inverse system of finite groups and let $G = \varprojlim G_i$ be its inverse limit, that is, a profinite group. If I know that all $\phi_{i,j}$ are surjective how can I conclude that the canonical projections $\phi_i: G \to G_i$ are surjective too?
My idea: If I have $x_i \in G_i$ then I can define the components of a $y \in G$ in following way:
For $i$: $y_i := x_i$; for j with $i>j$: $y_j := \phi_{i,j}(x_i)$. If $I$ is countable I can for every $k>i$ pull back the components $y_k$ of $y$ in following way by induction: $\phi_{k,i}$ is surjective so I can find a $x_k \in G_k$ with $\phi_{k,i}(x_k) = X_i$. So define $y_k:= x_k$. Because is a inverse system we get in this way an $y \in G$. But how can I pull back the components $y_k$ of y for $k>i$ to get a $y \in G$ with $\phi_i(y)= X_i$ if $I$ isn't countable?