I am searching for a proof that the Cech cohomology with values in locally constant functions, $\check{H}^p(\mathcal{U}, \mathbb{R})$, for a good open cover of the space, $X$ (with whatever reasonably nice conditions we usually want to impose on the space), is isomorphic to the direct limit over all open covers. I know the usual proofs which appeal to sheaf cohomology or DeRham cohomology, but I want a proof which sticks to the sheaf I am working with and is very direct: i.e. uses homological algebra and topology without appealing to partitions of unity, etc.
For starters, if anyone knows such a proof for: $\check{H}^p(V, \mathbb{R})= 0$ for $p>0$ and $V$ is a contractible space. Again, without appealing to hypercohomology, general sheaf cohomology methods (stalks, resolutions), or partitions of unity.
If you're curious why I'm being so stubborn about these constraints: I want to know if it's possible to prove it directly for my own curiosity but also so that I can share these ideas with advanced undergraduate students.