I'm trying to understand an example (A.64) in the appendix of Milne's 2017 book on algebraic groups. It goes like this.
Let $K/k$ be a Galois extension with Galois group $\Gamma$. We want to show that the functor $V \rightsquigarrow V \otimes_k K$ is an equivalence of categories from $k$-vector spaces to $K$-vector spaces with continuous semilinear $\Gamma$-action.
I understand the argument that Milne gives in the case where $K/k$ is a finite extension. However, right before that, he says, "It suffices to prove this with $K$ a finite Galois extension." I do not understand why this is sufficient. I have spent some time trying to derive the infinite case from the finite case, and am stuck.
I feel like it should be a somewhat straightforward consequence of the Galois correspondence for infinite extensions and the fact that the infinite Galois group is the inverse limit of intermediate Galois groups, but I can't figure it out.