Let $a,b$ be two natural numbers such that $a\leq b$, and let $P(n)$ denote some statement such that the following conjunction holds:
$$P(b)\ \land\ (\forall n)(n\in \mathbb{N}\ \land\ a\leq n<b\ \land\ P(n+1)\implies P(n)).$$
Then, $(\forall n)(n\in\mathbb{N}\ \land\ a\leq n\leq b\implies P(n))$.
Are there variants of this in which one can replace $P(n+1)$ in the inductive hypothesis with something like $P(m)$ for all $m>n$? And, generally speaking, can this technique only be carried out over some finite interval of $\mathbb{N}$?
Thank you in advance.