I found that we can prove $(m \mid n \iff F_m \mid F_n)$ by using matrix formulation or general formula for $F_n$ (Does $\,m\mid n$ iff $ F_m\mid F_n$?). But I'm just exposed to set theory, Peano's axioms, addition and multiplication.
I would like to know if we can prove that theorem just by using Principle of Strong Induction. I got stuck at the stage on which I have to prove that $P(k+1)$ is true.
Below is my attempt:
Let $P(n)$ is the statement $\forall m \in \mathbb{N}(m \mid n \iff F_m \mid F_n)$.
It is clear that $P(0)$ is true.
Assuming that $P(k)$ is true for all $k \leq n$ i.e. $\forall k \leq n[\forall m \in \mathbb{N}(m \mid k \iff F_m \mid F_k)]$.
We now have to prove $P(n+1)$ is true i.e. $\forall m \in \mathbb{N}(m \mid n+1 \iff F_m \mid F_{n+1})$.
From this point, I have no idea how to go on.
Please shed me some light! Thank you for your help ^_^
