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?

If a sequence is given by $S_{n+1}=S_n + S_{n-1}$ where $S_1=1$ and $S_2=2$ and we let $w_n=S_n/S_{n-1}$ for all $n \geq 2$. I trying to prove that for each $m \geq 2$, $|w_{m+1}-w_m| \leq (2/3)^{2(m-1)}|w_2 - w_1|$, but I don't understand how. I think I need to start with $|w_{m+1}-w_m| =$ so...

In the past we had a couple of discussions, and it was decided -- I thought -- that the tag proof shouldn't be used. Now we have more than 50 questions with this tag again. Could we please get it merged with make it synonym of [tag-removed]? (Or disable it in some other way.)

There are many tags which are blacklisted. And so are some strings/words/phrases in the title. Is it possible to have a list of them publicly available somewhere? As I was told by a moderator in chat: "Some of the blacklisted input is more sensitive (some stuff that helps prevent certain type...