Suppose $U'\cup U''=X$ is an open cover $U$ of a topological space $X$ and $F$ is a sheaf on $X$ with values in abelian groups. There is a special instance of the Grothendieck spectral sequence relating Cech to sheaf cohomology:

$$E_2^{p,q}=\tilde{H}^p(U,H^q(-,F))\Rightarrow H^{p+q}(X,F)$$

I would like to see, how this implies the Mayer-Vietoris sequence for this easy cover $U$. Drawing the $E_2$-page, I get so far that only the first two columns $p=0,1$ are non-zero. Therefore this page equals the $E_\infty$-page.