Homogenization of an affine variety in projective space I think I have a fundamental understanding issue. I am trying to prove that if we embed $\mathbb{A}^{n}$ into $\mathbb{P}^{n}$ by $x_{0} \neq 0$, and we have affine algebraic sets $V(I) \subset V(J)$, then $V(I^{*}) \subset V(J^{*})$, where the star denotes the homogenization of the ideals I and J i.e. homogenize a finite set of generators. $V(I^{*}) \cap U_{0} \subset V(J^{*}) \cap U_{0}$ is obvious, where $U_{0}$ is defined by $x_0\neq 0$. However, I can't figure out what happens when $x_0 = 0$. If $p \in I$ then $p^{*} = x_0^{d}p_{0} +.+ x_0^{i}p_{d-i} +..+ p_{d}$ where $p_{i}$ are the homogeneous components of $p$. So, at the hyperplane at infinity, I need to compare the roots of $p_{d}$ to the roots of $h_{k}$, where d and k are the degrees of $p \in I$ and $h \in J$. I tried to look it up, but a note online said that the result is obvious, so I must be missing something.. Thanks a lot

Hint:

We may as well assume that $I$ and $J$ are radical (why?). Note then that $V(I)\subseteq V(J)$ is equivalent to $I\supseteq J$. Clearly then $I^\ast\supseteq J^\ast$, and so $V(I^\ast)\subseteq V(J^\ast)$.