[proofplan] The first assertion is Cartan's Theorem B. The Leray assertion follows because finite intersections in a Stein cover are Stein in the setting used for the course, so Theorem B kills higher cohomology on all intersections. Exactness of global sections follows by applying the long exact cohomology sequence to short exact sequences of coherent sheaves. [/proofplan] [step:Apply Cartan's Theorem B] Since $X$ is Stein and $\mathcal{F}$ is coherent, Cartan's Theorem B gives \begin{align*} H^q(X,\mathcal{F})=0 \qquad (q\ge 1). \end{align*} This proves the acyclicity assertion. [/step] [step:Show Stein covers are Leray covers] Let $\mathcal{U}=\{U_i\}$ be a cover by Stein open sets such that finite intersections appearing in the Čech complex are Stein. For every finite intersection $U_{i_0\cdots i_r}$, the restriction $\mathcal{F}|_{U_{i_0\cdots i_r}}$ is coherent. Cartan's Theorem B gives \begin{align*} H^q(U_{i_0\cdots i_r},\mathcal{F})=0\qquad(q\ge 1). \end{align*} Thus the cover is $\mathcal{F}$-acyclic. Leray's acyclicity theorem then identifies Čech cohomology for this cover with sheaf cohomology. [/step] [step:Deduce exactness of global sections] Let \begin{align*} 0\to\mathcal{F}'\to\mathcal{F}\to\mathcal{F}''\to 0 \end{align*} be a short exact sequence of coherent sheaves on $X$. The associated long exact sequence begins \begin{align*} 0\to H^0(X,\mathcal{F}')\to H^0(X,\mathcal{F})\to H^0(X,\mathcal{F}'')\to H^1(X,\mathcal{F}')\to\cdots. \end{align*} By Theorem B, $H^1(X,\mathcal{F}')=0$. Therefore \begin{align*} 0\to H^0(X,\mathcal{F}')\to H^0(X,\mathcal{F})\to H^0(X,\mathcal{F}'')\to 0 \end{align*} is exact. Hence the global section functor is exact on coherent sheaves over $X$. [/step]