in a first semester exercice-group I'm tutoring students were given the standardproblem to prove that for $a>0$ $$a+\frac{1}{a}\geq 2 (*)$$ which is of course easy by multiplying with a and using binomials theorem.
However one of the students tried induction on $a$ which of course went horribly wrong but got me thinking. Maybe we can prove the formula by induction on $a$ for all $n\in\mathbb{N}$ which also would mean we have proved it for all $\frac{1}{n}$. If we then can get a additive or multiplicative Identity (if $(*)$ holds for $a$ and $b$ then it also holds for $a+b$, or maybe for $a*b$). This would give us the statement for all $a\in\mathbb{Q}$ and then we could use density.
But until now I didn't find a way to do this (neither the induction nor ne addition/multiplication part) and I'm not sure if it can be done. Are there any ideas?