I don't understand a step in this proof of the Hilbert Basis Theorem. Here is the proof Planeth Math.
I don't understand why $ \mathrm{deg} (f_{N+1}-g)< \mathrm{deg}(f_{N+1}) $. This can only happen if $ g $ has the same degree as $ f_{N+1} $ and also the same leading term, but I don't know why.