I am looking at exercise 2.44 in William Fulton's "Algebraic Curves". It asks to prove that
Let $V$ be a variety in $\mathbb{A^n}$, $I = I (V ) \subset k[X1, . . . ,Xn], P \in V $, and let $J$ be an ideal of $k[X1, . . . ,Xn]$ that contains $I$. Let $J'$ be the image of $J$ in $\Gamma(V )$. Show that there is a natural homomorphism $\gamma$ from $ O_p (\mathbb{A^n})/JO_p (\mathbb{An})$ to $O_p (V )/J'O_p (V )$, and that $\gamma$ is an isomorphism. In particular, $O_p (\mathbb{A^n})/IO_p (\mathbb{An})$ is isomorphic to $O_p(V)$.
I was thinking about it but I do not know how to start. Somebody can help me and give me a hint.
Let's start by trying to define a homomorphism $O_p(\Bbb A^n)\to O_p(V)$. An element of $O_p(\Bbb A^n)$ is of the form $f/g$ with $f,g\in\Gamma(\Bbb A^n)$ with $g(p)\neq0$. If we take $\bar f:=f\:(\mathrm{mod}\:I)$ and similarly define $\bar g$, then $\bar f,\bar g\in\Gamma(V)$ and we have $\bar g(p)\neq0$ (be sure of each of these things!). Therefore we can define $O_p(\Bbb A^n)\to O_p(V)$ by $f/g\mapsto \bar f/\bar g$ (again, you should be sure this is well-defined!)
Now we can take the composition $O_p(\Bbb A^n)\to O_p(V)\to O_p(V)/J' O_p(V)$. Is this surjective? (Hint: the map we defined in the above paragraph is surjective). What is the kernel?