Let $A$ be an integrally closed domain of Krull dimension 1, $K$ its fraction field, $L|K$ a finite or infinite Galois extension and $B$ the integral closure of $A$ in $L$. Also, let $\mathfrak{P}\subset B$ be a non zero prime ideal of $B$ and $\mathfrak{p}:=\mathfrak{P}\cap A$ its contraction to $A$. I must prove that the following properties hold:
The residual extension $(B/\mathfrak{P})|(A/\mathfrak{p})$ is a normal algebraic extension of fields.
The decomposition and inertia groups, $G_{-1}(\mathfrak{P}|\mathfrak{p})$ and $G_{0}(\mathfrak{P}|\mathfrak{p})$, are closed subgroups of $\text{Gal}(L|K)$ with respect to the Krull topology.
I've been able to prove both properties in the finite case, but I have the feeling that in order to prove them in the infinite case I will need projective limits, which is a concept I'm not familiarized with. I'm also assuming that $L$ can be written as a countable union of fields $K_n$, which are finite Galois extensions of K such that $K_n\subset K_{n+1}$, that is, $\{K_n\}_{n\in\mathbb{N}}$ is an increasing sequence of fields.
How should I proceed?
You are correct that you will need profinite constructions, but it's actually quite easy: by definition something is true in the limit if it is true at every finite step. So all you need to do is note that the Galois groups of the infinite extension is a limit over all finite sub-extensions, which is true by construction. Next verify they form a projective system--which is easy, the connecting maps are just the projections onto the smaller Galois groups for finite extensions (which exist by the fundamental theorem of Galois theory). Then by definition the normality condition holds in the limit because it does for every finite, Galois sub-extension.
For part (b) write the decomposition group for the big extension as the intersection of decomposition groups for finite extensions which are already known to be closed subgroups. Since the intersection of closed sets are closed, it follows here as well.