# Mobius transformation maps $\Bbb R_∞$ onto itself iff we can choose its coefficients to be real

I have seen many solutions which are very intricate and/or long. The solution I had was much shorter which lead me to believe that it was incorrect. It went like this :

$$T$$ be a Mobius transformation such that $$T(\Bbb R_∞)$$=$$\Bbb R_∞$$. $$1,0,∞\in \Bbb R_∞$$ so their inverse images are also in $$\Bbb R_∞$$. $$(z,z_2,z_3,z_4)$$ is the unique Mobius transformation that maps $$z_2,z_3,z_4$$ to respectively $$1,0,∞$$. So when we write $$T$$ as a cross-ratio, the coefficients are all real.

Is this solution really incorrect?

I think that I'd edit slightly by saying "so their inverse images, $$\mathbf{ z_2, z_3, z_4}$$ are also in..." so that $$z_2, z_3, z_4$$ are actually defined. Otherwise it looks good to me.