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 $\pm\infty$?

It is meaningless to say that "$n$ takes $\pm \infty$" so long as $n\in \mathbb{N}$ simply because $\pm \infty \not \in \mathbb{N}$. So if $n$ is finite then the Induction Principle says that the integer $n$ can be reached starting from $1$ (if the Induction Basis exists) in finitely many steps, however large. Hence one claims that for all $n\in \mathbb{N}$ the proposition holds, which as a corollary gives us the conclusion that the proposition holds for all finite $n$ since any finite $n$ is a member of $\mathbb{N}$. 
