You can prove it through mathematical induction. Show that if it's valid for n dimensions then it's valid for n+1 dimensions. So then if it's proven for R_2 it's proven in general.
The induction in question here is of course on the dimension of the vector space, which is a natural number -- not the members of the vector space itself.
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 .