No need to start with R^2; start with R^1, where it's easy. Starting the proof with R^2 essentially means that you must do the induction step twice. For an am

No need to start with R^2; start with R^1, where it's easy. Starting the proof with R^2 essentially means that you must do the induction step twice. For an amusing instance of induction on dimension, you might enjoy the proof of the AMGM inequality, which proceeds by upwards induction that doubles the dimension, followed by downward induction: https://proofwiki.org/wiki/Cauchy's_Mean_Theorem#Theorem https://proofwiki.org/wiki/Cauchy's_Mean_Theorem#Theorem .