I am trying to do exercise 11.3.c part b from Vakil's note. It asks to show that if $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension $r$ and $Y$ is a closed subset of codimension $r$, then they always meet in some point. Following the solution I found here Trouble with Vakil's FOAG exercises 11.3.C I tried to proceed as follows:
Let X = V(I) be a closed subset of $\mathbb{P}^{n}_{k}$ of dimension $r$ and $Y = V(J)$ be a closed subset of codimension $n-r$. I assumed, even though Vakil doesn't say it, that $Y$ is of pure codimension $r$, otherwise I think the exercise could be not true. Please correct me if I'm wrong. Up to take an irreducible component of $Y$, we may assume $J$ is a prime ideal. Then, as $Y$ has codimension $r$, $J$ has codimension $r$, and therefore it is minimal over an ideal generated by some elements $a_1, \dots, a_r$. Let us denote $\mathfrak{q} = (a_1, \dots, a_n)$. Then, we have $I + \mathfrak{q} \subset I + J \subset (X_0, \dots, X_n)$ and therefore there exists a minimal prime ideal $\mathfrak{p}$ such that $I + \mathfrak{q} \subset \mathfrak{p} \subset ( X_0, \dots, X_n)$. Now proceeding as in Trouble with Vakil's FOAG exercises 11.3.C we can prove that $\mathfrak{p} \subsetneq (X_0, \dots, X_n)$. However, from here I don't know how to deduce that there exists at least one of these $\mathfrak{p}$ such that $I + J \subset \mathfrak{p}$.