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 $\pm\infty$?

Mathematical induction is based on two properties of the natural numbers.  They are the Well-Ordering property (each non-empty subset of the natural numbers has a least element) and the fact that each element has a successor.
If we wish to establish that a proposition $P(n)$ is true for each $n \in \mathbb{N}$ such that $n \geq n_0$, we must first establish that $P(n_0)$ holds, which establishes that the set $S = \{n \in \mathbb{N}|~P(n)~\text{holds}\} \neq \emptyset$.  We then show that $P(n) \Rightarrow P(n + 1)$, which demonstrates that if $n \in S$, then its successor $n + 1 \in S$.  Since $n_0 \in S$, the fact that $P(n) \Rightarrow P(n + 1)$ for each $n \in S$ allows us to conclude that $n_0, n_0 + 1, n_0 + 2, \ldots \in S$.  Therefore, $S = \{n \in \mathbb{N}|~n \geq n_0\}$.  However, it does not demonstrate that $\infty \in S$ since $\infty$ is not the successor of any natural number. 
