Is $n\in\mathbb{Z^{+}}$ or $n\in\mathbb{N}$ more appropriate for proof by induction?

Question

Some textbooks say that proof by mathematical induction is applicable when $n$ is a positive integer, while others say $n\in\mathbb{N}$. Which one of ‘em is correct tho? (since set of positive integers doesn't include 0)

Answer 1

As far as the general structure of an arbitrary induction proof goes, the only thing you're doing with the different $n$ is comparing them (you're not adding or multiplying them or something like that). And in that respect, the two sets (natural numbers with $0$ or without $0$) are essentially indistinguishable. You can use whichever suits your purposes best.

Also note that it is not universally assumed that $0\in \Bbb N$. I'm not even consistent myself.