 Solution

For $X=U\cup V$, the Mayer-Vietoris sequence for sheaf cohomology is the long exact sequence

We prove the required vanishing by induction on the number $m$ of open sets. The case $m=1$ is an assumption. Put $U=U_1\cup \cdots \cup U_{m-1}$ and $V=U_m$. The induction hypothesis gives $H^p(U,\mathcal{F})=0$ for every $p$. The intersections $U_i\cap U_m$ cover $U\cap V$, and every nonempty finite intersection among them is one of the intersections in the hypothesis, so the same induction gives $H^p(U\cap V,\mathcal{F})=0$. We also have $H^p(V,\mathcal{F})=0$. Exactness of the Mayer-Vietoris sequence now yields