How can one show that rational functions satisfy a Lipschitz condition in the chordal metric on the Riemann sphere?
By compactness, you only need to prove the Lipschitz property locally. First prove that Möbius transforms are Lipschitz (easy – they are compositions of translations, multiplications by constants, and inversions). Then, by composing with suitable Möbius transforms, you only need to show that a rational function which maps 0 to 0 is Lipshitz on a neighbourhood of 0. This is trivial, I hope you agree.