No need to start with R^2; start with R^1, where it&#x27;s easy.  Starting the proof with R^2 essentially means that you must do the induction step twice.<p>For an amusing instance of induction on dimension, you might enjoy the proof of the AMGM inequality, which proceeds by upwards induction that <i>doubles</i> the dimension, followed by downward induction:  <a href="https:&#x2F;&#x2F;proofwiki.org&#x2F;wiki&#x2F;Cauchy&#x27;s_Mean_Theorem#Theorem" rel="nofollow">https:&#x2F;&#x2F;proofwiki.org&#x2F;wiki&#x2F;Cauchy&#x27;s_Mean_Theorem#Theorem</a> .