Absolute Galois group and continuous homomorphism. Let $G=\mathrm{Gal}(\bar K/K)$ be the absolute Galois group of a number field $K$.
I have some problem to understand that the homomoprhism
$$ \rho :G \longrightarrow \mathrm{GL}_r(E)$$
$E$ a finite extension of $\mathbb{Q}_{\ell}$, is continuous if and only if for all $n$ there exist a finite extension $E_n/K$  such that $\mathrm{Gal}(\bar K/E_n) \mapsto \mathrm{id} \ \ (\mathrm{mod}\: \ell^n)$. And what about if I replace $E$ with $\mathbb{Z}_{\ell}$ or $\mathbb{Z}/\ell \mathbb{Z}$?
 A: I think part of the issue is that your question doesn't make sense literally, unless you are committing an identification which, if you understood, would probably make your question moot.
In particular, what does it mean to reduce a matrix in $\mathrm{GL}_r(E)$ modulo $\ell^n$? The thing you are secretly assuming, which is fine, is that by continuity (and compactness of $G$) $\rho$ stabilizes a $\mathcal{O}_E$-lattice in $E^r$. Thus, up to changing basis, we may as well assume that we are given $\rho$ as a continuous morphism $G\to\mathrm{GL}_r(\mathcal{O}_E)$. But, $\mathrm{GL}_r(\mathcal{O}_E)$ is profinite. In particular:
$$\mathrm{GL}_r(\mathcal{O}_E)=\varprojlim \mathrm{GL}_r(\mathcal{O}_E/\pi^n)$$
where $\pi$ is a uniformizer of $E$. So, we see that $\rho:G\to\mathrm{GL}_r(\mathcal{O}_E)$ is continuous with the $\pi$-adic topology if and only if the maps $\rho_n:G\to\mathrm{GL}_r(\mathcal{O}_E/\pi^n)$ are continuous with the discrete topology for all $n$.
But, by definition this means that $\ker\rho_n$ is open. Thus, we are asking that for all $n$ there is a finite extension $E_n$ of $K$ such that 
$$\rho_n(G_{E_n})=\{I_r\}\subseteq\mathrm{GL}_r(\mathcal{O}_E/\pi^n)$$
Now, this answers your question with $\mod \ell^n$ replaced with $\mod \pi^n$. But, since $E/\mathbb{Q}_\ell$, so in particular $\mathcal{O}_E\supseteq\mathbb{Z}_\ell$, it's not hard to see how to finish the argument (Hint: consider the canonical map $\mathrm{GL}_r(\mathcal{O}_E/\ell^n)\to\mathrm{GL}_r(\mathcal{O}_E/\pi^n)$)
