Question on induction technique When one uses induction (say on $n$) to prove something, does it mean the proof holds for all finite values of $n$ or does it always hold when even $n$ takes $\pm\infty$?

Regular inductive proofs work only over $\mathbb{N}$, the set of natural numbers. The proof that "inductive proofs work" depends on the well-ordering property of $\mathbb{N}$, and $\infty \not\in \mathbb{N}$. As people in the comments have pointed out, induction can be extended to other well-ordered sets as well. Regardless, it is meaningless to ask if a statement holds at $\infty$.
