I cannot seem to solve what I think should be a straightforward intersection theory question in Ravi Vakil's algebraic geometry notes.
Let $X$ be a scheme and $Y$ a closed subscheme of dimension less than or equal to $n$, and $\mathscr{L}_1,\ldots,\mathscr{L}_n$ be a collection of line bundles on $X$. Then Vakil defines the intersection product:
$$(\mathscr{L}_1 \cdots \mathscr{L}_n\cdot Y)=\sum_{\{i_1,\ldots,i_m\}\subseteq \{1,\ldots,n\}}\chi(X,\mathscr{L}_{i_1}^\vee\otimes\cdots \otimes \mathscr{L}_{i_m}^\vee \otimes \mathscr{O}_Y)$$ where $\chi$ is the euler characteristic and $\mathscr{O}_Y$ is really its pushforward to $X$.
Problem 20.1.B asks if $k$ is field and $X=\mathbb{P}^n_k$ and $Y$ any dimension $n$ subscheme, $\{H_i\}_{i=1}^n$ hypersurfaces of degree $d_i$, then we are asked to show $$(\mathscr{O}(H_1) \cdots \mathscr{O}(H_n)\cdot Y)=d_1\cdots d_n \operatorname{deg}Y$$
I am having a hard time on this, because I cannot seem to find a natural way to induct. I think there should be a way to reduce the problem to $Y\cap H_1$ and then induct, but I cannot figure out the details on how to do that.
Any help or direction would be much appreciated.