Prove that a Möbius Transformation maps the real axis to the real axis iff the coefficients of the Möbius Transformation are real. If I assume the coefficients are real and take the $3$ points on the real line and show that it maps to $3$ points on the real line I think I can show the converse.
So,
If we have the Möbius Transformation $T(z)=\frac{az+b}{cz+d}$ and I assume $a,b,c,d$ are real, then if I take the points, $-1,1,\infty$ and show it maps to $3$ real points via the transformation I would think that is sufficient. Noting $T(\infty)=\frac{a}{c}$ (which is real if coefficients are real).
How would I go about proving the $\to$ direction?
Sorry, not too familiar with Latex.