Proving that a variety of dimension zero is discrete. Say we have a variety of dimension zero, how do we prove this is discrete?
I have some ghost of an idea of what is going on, thanks to threads like this 
Why is every Noetherian zero-dimensional scheme finite discrete?
but I cant formulate a concrete proof, even after reducing it to the affine case. 

A noetherian ring has only finitely many minimal prime ideals. In a zero-dimensional ring, any prime ideal is maximal. Combining these two statements, we get that the spectrum of a noetherian ring of dimension zero is a finite set of closed points. In particular, the spectrum is discrete since any sub-set is closed (being a finite union of closed subsets).
