Examples of certain proofs by induction

Question

What are the best examples of proof by mathematical induction on $n$ of statements of the form $\left( \forall n \in \{1,\ldots, N\}\,\,\Big(\cdots\cdots\cdots\Big) \right),$ where there's some good reason why there are only finitely many cases?

Answer 1

There's the following unusual proof of (more or less) Fermat's little theorem: for all $x \in \mathbb F_p$, $x^p - x = 0$.

For $x=0$, we have $0^p - 0 = 0$. Assume that for some $x$, $x^p-x = 0$, and consider $(x+1)^p - (x+1)$. Because we're working in characteristic $p$, $(x+1)^p = x^p + 1$; therefore $$(x+1)^p - (x+1) = x^p + 1 - x - 1 = x^p - x = 0.$$ By induction, this claim holds for all $p$ values of $x$.