= Rational map is Lipschitz in the chordal metric
{title2=$\chi(Rz,Rw)\leq C_R\chi(z,w)$}
A rational map is smooth as a map between compact round spheres, including at its poles in reciprocal coordinates. Its spherical differential norm therefore has a finite supremum $L$. Integrating along a shortest spherical geodesic gives $\rho(Rz,Rw)\leq L\rho(z,w)$. Since $2\chi\leq\rho\leq\pi\chi$, we obtain $\chi(Rz,Rw)\leq(\pi L/2)\chi(z,w)$. In finite coordinates the differential norm is $R^\#(z)=|R'(z)|(1+|z|^2)/(1+|R(z)|^2)$, extending continuously to the whole sphere. This uniform bound is useful for passing equicontinuity through compositions.
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