Let $G_\mathbf{Q}$ denote the absolute Galois group of $\mathbf{Q}$. I'm trying to understand the proof that image of any continuous homomorphism $G_\mathbf{Q}\rightarrow\mathrm{GL}_n(\overline{\mathbf{Q}}_p)$ has image in $\mathrm{GL}_n(E)$ for some field $E$ finite dimension over $\mathbf{Q}_p$. The proof uses Baire theorem and elsewhere on MSE someone said it works with $G_\mathbf{Q}$ replaced by any profinite group. Here is the proof in notes by Conti:
The part I don't understand has the ???. How are we applying Baire's theorem to get the result? My guess was that for some reason $\rho(G)$ is not nowhere dense - meaning its closure (it's already closed by the way) contains an open set, but I couldn't figure out why that must be true - and the trivial representation shows it is not true (but the trivial representation takes values in $\mathbf{Q}_p$ already, of course).
