TITLE: What are statements about the natural numbers where induction is impossible or unnecessary to prove?
QUESTION [14 upvotes]: I'm looking for statements like "for all natural numbers, ____" where induction would be impossible or unnecessarily complicated. This is for pedagogical reasons. When students first learn induction, they want to prove every statement about natural numbers by induction, even when it is entirely not necessary or possible. It would be nice to have examples to break "natural numbers implies induction".
For example: for all $n\in \mathbb{N}$, $\int_0^{2\pi} \cos(nx) dx=0$. If one tries to show this by induction, it would be way more complicated than necessary. You can just compute the integral directly (assuming you know $\sin(2n\pi)=0$).

REPLY [2 votes]: Start with the list of all $2^n$ vectors of length $n$ of $+1$s and $-1$s. Someone changes some of the entries to $0$. Show that there is always a non-empty collection of rows which sums to the zero vector.
This looks like the perfect case for induction (on $n$), but the two proofs I'm aware of don't use induction; induction just doesn't seem to help here.