I am talking about the theorem from Galois theory, specifically to Milne's course notes on field theory proposition 7.10 (which can be easily found online).
In the proof of this proposition, we have a a map of topological groups $G\to Gal(E/E^G)$, which is injective, with compact image, and maps onto every subset of the form $Gal(M/E^G)$ where $M$ is some finite Galois extension of $E^G$. From this we deduce first that the image is dense, and then that the map is in fact surjective. How do we accomplish these two steps?
Thank you very much