The point-divisor exact sequence and the Euler-characteristic step

Fix the disk $(W,z)$ from the statement, small enough to meet no support point of $D$ other than possibly $p$. For a germ $f$ in ${\mathcal{O}}_X(D+p{)}_p$, [F3] gives ${\mathrm{ord}}_p(f)\ge -k-1$, so its Laurent expansion on a sufficiently small punctured disk has only powers $z^n$ with $n\ge -k-1$ by [F4]. Define ${\lambda }_{p,z}$ on a section over any open containing $p$ by taking this germ coefficient $c_{-k-1}$, and define it to be zero on opens not containing $p$. Uniqueness of Laurent coefficients makes this independent of the smaller disk used to compute it, and restrictions preserve the coefficient, so these maps form a sheaf morphism. Its kernel at $p$ consists exactly of germs with order at least $-k$, which is ${\mathcal{O}}_X(D{)}_p$; it is surjective at $p$ because the germ $z^{-k-1}$ maps to $1$. At any $x\mathrm{\ne }p$, the divisors $D$ and $D+p$ agree near $x$, so the inclusion is an isomorphism on that stalk and $({\mathbb{C}}_p{)}_x=0$. Thus the stalk sequence is exact at every point, and [F5] gives the asserted short exact sequence.

For open sets $V\subseteq U$, the restriction ${\mathbb{C}}_p(U)\to {\mathbb{C}}_p(V)$ is the identity if both contain $p$ and is the zero map to $0$ otherwise, so it is always surjective by [F6]. Hence ${\mathbb{C}}_p$ is flasque. Since $p\in X$, its global sections are $\mathbb{C}$. Applying [F7] to the flasque sheaf on $X$ gives $H^q(X,{\mathbb{C}}_p)=0$ for every $q\ge 1$.

Apply [F8] to the short exact sequence in step 1.1 and use step 1.2; the relevant portion is the displayed six-term exact sequence, with final zero because $H^1(X,{\mathbb{C}}_p)=0$. Choose compatible metrics on $X$, ${\mathcal{O}}_X(D)$ and ${\mathcal{O}}_X(D+p)$ using [F9] and the countable instances of full AC in [F1]. Applying the finiteness theorem in [F9] with these metrics makes the $H^0$ and $H^1$ terms for both divisors finite-dimensional, and $\dim H^0(X,{\mathbb{C}}_p)=1$. Exactness gives $\dim \mathrm{im}{\lambda }_{p,z}=\mathrm{\ell }(D+p)-\mathrm{\ell }(D)$ and makes $H^1(X,{\mathcal{O}}_X(D))\to H^1(X,{\mathcal{O}}_X(D+p))$ surjective. If $r=\dim \mathrm{im}{\lambda }_{p,z}$, exactness also gives $i(D)-i(D+p)=1-r$; hence $\chi ({\mathcal{O}}_X(D+p))-\chi ({\mathcal{O}}_X(D))=(\mathrm{\ell }(D+p)-\mathrm{\ell }(D))+(i(D)-i(D+p))=r+(1-r)=1$.