I was trying to solve the following exercise of elliptic curves:
Let $F$ be an isogeny such that there exists $n,m\in\mathbb{N}$ with $[n]-[m]F=[0]$, then $F=\hat{F}$, where $\hat{F}$ is the dual isogeny.
At first sight it looks a trivial exercise, but I got stuck at the last part. This is my attempt:
Taking dual isogenies: $$[n]-\widehat{F}[m]=\widehat{[n]}-\widehat{F}\widehat{[m]}=\widehat{[n]-[m]F}=\widehat{[0]}=[0]=[n]-[m]F$$ Here I only used basic properties of isogenies. Thus we have $\widehat{F}[m]=[m]F$. How am I supposed to show that $F=\widehat{F}$ from here? I guess that the exercise is meant to be solved applying elementary properties of rational maps and dual isogenies, but I don't see what I'm missing.
Any hints or help will be appreciated.