All authors I have seen handwave the proof of the Čech complex actually being a complex as a calculation. But I tried to do it and I don't see how it works. Maybe there's a mistake in my calculation.
For a sheaf $F$ and the cover $\{ U_i \}$, I want $C^p \xrightarrow{d} C^{p+1} \xrightarrow{d} C^{p+2}$, $d \circ d = 0$. To calculate, \begin{align} (d(ds))_{i_0, \ldots, i_{p+2}} &= \sum\limits_{k=0}^{p+2} (-1)^k (ds)_{i_0, \ldots, \hat{i_k}, \ldots, i_{p+2}} \rvert_{U_{i_0, \ldots, i_{p+2}}} \\ &= \sum\limits_{k=0}^{p+2} (-1)^k \left.\left( \sum\limits_{\substack{l=0 \\ l\neq k}}^{p+2} (-1)^l s_{i_0, \ldots, \hat{i_l}, \ldots, \hat{i_k}, \ldots, i_{p+2}} \rvert_{U_{i_0, \ldots, \hat{i_k}, \ldots, i_{p+2}}} \right) \right\rvert_{U_{i_0, \ldots, i_{p+2}}} \\ &= \sum\limits_{\substack{k, l = 0 \\ k \neq l}}^{p+2} (-1)^{k+l} s_{i_0, \ldots, \hat{i_l}, \ldots, \hat{i_k}, \ldots, i_{p+2}} \rvert_{U_{i_0, \ldots, i_{p+2}}} \end{align}
The last equality uses that $U_{i_0, \ldots, i_{p+2}} \subseteq U_{i_0, \ldots, \hat{i_k}, \ldots, i_{p+2}}$. How can I see from here that this sum reduces to $0$?