in another question (or in Wedhorn's Algebraic Geometry, page 88) it is shown that a subscheme of $k$-schemes of finite type are of finite type again.
Does the same hold closed subschemes of finite $k$-schemes?
Remark: What I really am trying to prove is that this statement holds for group schemes. Here, the closed subschemes are given by quotients $A/I$ of a Hopf algebra $A$ by an Hopf ideal $I$ and one needs to prove that this is finite-dimensional as a $k$-vector space. However, I don't think I can simply think of $A/I$ as a quotient vector space.
Any help would be appreciated!