I'm reading and trying to understand the proof of the finiteness of n-Selmer group from J.S.Milne's Elliptic Curves book but having difficulty in understanding it. Here's a screenshot from the book-

Now, what I don't understand in this proof is- since we want to show the exactness of a sequence it suffices to prove it at $N$, but I don't understand how the proof in the book implies it.

Also, how can I make sense of the map $[n]$ from $C_{T}$ to $C_{T}$?

I would appreciate if it if someone could dumb it down (the proof) or maybe could suggest some reference where the intermediate steps are done.

Thank you!

EDIT: Also, I would like it very much if someone could suggest some references where I can the theory related to the finite extensions of $L_v$, where $L$ is a number field and $L_v$ is it's completion with respect to a valuation. It's making me very frustrated unable to fill the gaps in the proof because I don't have the knowledge of some pre requisite algebra. I'd appreciate any help.