I don't understand the proof of Noether Normalization Lemma in "Algebraic Geometry and Arithmetic Curves" . Liu considers first the case $k[X_1,X_0]/I$, then $k[X_1,X_1]/I=k[X_1]/I$, then again $k[X_1,X_2]/I$. It feels strange that induction proves the theorem for two variables, then for one variable, then again for two variable and the rest cases. Is there a mistake in subscripts or does $k[X_1,X_0]/I=k/I$?
It looks like a mistake or then I have misunderstood the notation of $k[X_1,X_0]$ as the proof given in http://www.proofwiki.org/wiki/Noether_Normalization_Lemma follows from induction on number of generators.