How do you go from "and so on 3,4,5,6,7,8,9,10,.........." to "So N is infinite"?

When you say "and so on 3,4,5,6,7,8,9,10,.........." you are simply saying that you can get all of N that way. Asserting that it is infinite is just assuming that N is infinite, which is what you are trying to prove.

By the way, whatdefinitionof "infinite" are you using?