What exactly is your question? How the proof works? What „Every infinite set is T-infinite“ means?
Regarding the latter: A set is S T-finite if for EVERY subset A of its power set P(S) there is a „setsize“- maximal element.
A set is T-infinite if it is not T-finite, that means there is a subset X of its power set P(S) such that there is no „setsize“-maximal element. So if S is T-infinite, there is subset if its power set with infinitey large elements :)