Please forgive me if this has already been asked(my initial searches didn't find anything close enough).
I am studying Mathematical Induction and in the Inductive Step when we Assume Statement True for Nth Element and prove it holds for (N+1)th Element, hence it holds for Entire Set<< EXACTLY here
Does this mean that we have just proven that the statement holds NOT ONLY for a Discreet Element(0) BUT ALSO for a PAIR of Arbitrary Elements bound by Successor Relation and since Starting from 0 we can make pairs of ALL Elements of a Set of Natural Numbers that are bound by Successor Relation, hence it holds for Entire Set...
Another add on question is; Can Induction be applied to prove theorems in ALL kinds of sets or there is a restriction like it can ONLY be applied on ordered sets, where elements of the set are in some kind of relationship with one another?
I am self-taught so my knowledge level is around high school I guess...
[EDIT:] It seems to me that since the inductive step proves that the statement holds for A PAIR of Consecutive Elements of the set AND since every consecutive pair has the EXACTLY same relationship with one another, the statement holds for ENTIRE SET. It seems to me that the Definition of the elements(with relations with one another) of the set plays a huge role in making Induction a valid proof. As mentioned by @DavidDiaz in comments that "Induction is for strictly well ordered set".