Intersection theory question from Vakil's notes on Algebraic Geometry 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.
 A: One way could be to use an induction technique similar to one needed for exercise 20.1.C.
Let me show 20.1.C in the case $n=2$; for higher dimension the proof becomes cluttered with indices.
Pull back effective Cartier divisor $D$ to $Y$ (avoiding associated points of $Y$). For this divisor $D \cap Y$ on $Y$ we have an exact sequence (14.3):$$0 \to \mathcal{O}_{Y}(-D\cap Y) \to \mathcal{O}_Y \to \mathcal{O}_{D \cap Y} \to 0 $$ 
We thus have $\chi(Y, \mathcal{O}_{D \cap Y})=\chi (Y, \mathcal{O}_Y) - \chi(Y, \mathcal{O}(-D \cap Y))$. Similarly, after tensoring the sequence with $\mathcal{L_1}^{\vee}|_Y$, one gets $\chi(Y, \mathcal{L_1}^{\vee}|_Y \otimes \mathcal{O}_{D \cap Y})=\chi(Y, \mathcal{L_1}^{\vee}|_Y \otimes \mathcal{O}_{Y})-\chi(Y, \mathcal{L_1}^{\vee}|_Y \otimes \mathcal{O}(- D \cap Y))$.
Next I will also use the fact that $\chi(Y, \mathcal{F}) = \chi(X, i_{*} \mathcal{F})$ for an affine morphism (e.g. closed embedding) $i: Y \to X$, but I will suppress this pushfordward throughout.
One then has: $$\mathcal{L_1} \cdot D \cap Y = \chi(\mathcal{O}_{D \cap Y}) - \chi(\mathcal{L_1}^{\vee}|_Y \otimes \mathcal{O}_{D \cap Y})=$$ $$=\chi ( \mathcal{O}_Y) - \chi(\mathcal{O}(-D \cap Y)) - \chi(\mathcal{L_1}^{\vee}|_Y \otimes \mathcal{O}_{Y})+\chi(\mathcal{L_1}^{\vee}|_Y \otimes \mathcal{O}(- D \cap Y))$$where we can identify second term as $\chi(\mathcal{O}(-D) \otimes \mathcal{O}_Y)$, and the last term as $\chi( \mathcal{L_1}^{\vee} \otimes \mathcal{O}(-D) \otimes \mathcal{O}_Y)$. Thus the whole expression is equal to $$\mathcal{L_1} \cdot \mathcal{O}(-D) \cdot Y$$as desired.
Now 20.1.B.b becomes quite easy - using 20.1.C inductively we get that $$\mathcal{O}(H_1) \cdot ... \cdot \mathcal{O}(H_n) \cdot Y = \mathcal{O}(H_1) \cdot (H_2 \cap H_3 \cap ... \cap H_n \cap Y)$$ Using the above exact sequence with $D=H_1$ and the definition of intersection product, the last expression becomes $$\chi(\mathcal{O}_{H_1 \cap ... \cap H_n \cap Y})=d_1 \cdot ... \cdot d_n \cdot deg(Y) \ \ \text{(by Bezout).}$$
