proof of Riemann-Roch theorem

For a divisor $D$, let $𝔏(D)$ be the associated line bundle. By Serre duality, $H^0(𝔏(K-D))\cong H^1(𝔏(D))$, so $\mathrm{\ell }(D)-\mathrm{\ell }(K-D)=\chi (D)$, the Euler characteristic of $𝔏(D)$. Now, let $p$ be a point of $C$, and consider the divisors $D$ and $D+p$. There is a natural injection $𝔏(D)\to 𝔏(D+p)$. This is an isomorphism anywhere away from $p$, so the quotient $ℰ$ is a skyscraper sheaf supported at $p$. Since skyscraper sheaves are flasque, they have trivial higher cohomology, and so $\chi (ℰ)=1$. Since Euler characteristics add along exact sequences (because of the long exact sequence in cohomology) $\chi (D+p)=\chi (D)+1$. Since $\mathrm{deg}(D+p)=\mathrm{deg}(D)+1$, we see that if Riemann-Roch holds for $D$, it holds for $D+p$, and vice-versa. Now, we need only confirm that the theorem holds for a single line bundle. ${𝒪}_X$ is a line bundle of degree 0. $\mathrm{\ell }(0)=1$ and $\mathrm{\ell }(K)=g$. Thus, Riemann-Roch holds here, and thus for all line bundles.