[Proof using spectral sequences.] Suppose that $\xi \in H^p(X, \mathcal{F})$ for some $p \geq 0$. Let us show that $\xi$ is in the image of the map $\check{H}^p(X, \mathcal{F}) \to H^p(X, \mathcal{F})$ of Lemma \ref{lemma-cech-spectral-sequence} for some hypercovering $K$ of $X$. \medskip\noindent This is true if $p = 0$ by Lemma \ref{lemma-h0-cech}. If $p = 1$, choose a {\v C}ech hypercovering $K$ of $X$ as in Example \ref{example-cech} starting with a covering $K_0 = \{U_i \to X\}$ in the site $\mathcal{C}$ such that $\xi|_{U_i} = 0$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}. It follows immediately from the spectral sequence in Lemma \ref{lemma-cech-spectral-sequence} that $\xi$ comes from an element of $\check{H}^1(K, \mathcal{F})$ in this case. In general, choose any hypercovering $K$ of $X$ such that $\xi$ maps to zero in $\underline{H}^p(\mathcal{F})(K_0)$ (using Example \ref{example-cech} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-kill-cohomology-class-on-covering} again). By the spectral sequence of Lemma \ref{lemma-cech-spectral-sequence} the obstruction for $\xi$ to come from an element of $\check{H}^p(K, \mathcal{F})$ is a sequence of elements $\xi_1, \ldots, \xi_{p - 1}$ with $\xi_q \in \check{H}^{p - q}(K, \underline{H}^q(\mathcal{F}))$ (more precisely the images of the $\xi_q$ in certain subquotients of these groups). \medskip\noindent We can inductively replace the hypercovering $K$ by refinements such that the obstructions $\xi_1, \ldots, \xi_{p - 1}$ restrict to zero (and not just the images in the subquotients -- so no subtlety here). Indeed, suppose we have already managed to reach the situation where $\xi_{q + 1}, \ldots, \xi_{p - 1}$ are zero. Note that $\xi_q \in \check{H}^{p - q}(K, \underline{H}^q(\mathcal{F}))$ is the class of some element $$ \tilde \xi_q \in \underline{H}^q(\mathcal{F})(K_{p - q}) = \prod H^q(U_i, \mathcal{F}) $$ if $K_{p - q} = \{U_i \to X\}_{i \in I}$. Let $\xi_{q, i}$ be the component of $\tilde \xi_q$ in $H^q(U_i, \mathcal{F})$. As $q \geq 1$ we can use Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-kill-cohomology-class-on-covering} yet again to choose coverings $\{U_{i, j} \to U_i\}$ of the site such that each restriction $\xi_{q, i}|_{U_{i, j}} = 0$. Consider the object $Z = \{U_{i, j} \to X\}$ of the category $\text{SR}(\mathcal{C}, X)$ and its obvious morphism $u : Z \to K_{p - q}$. It is clear that $u$ is a covering, see Definition \ref{definition-covering-SR}. By Lemma \ref{lemma-covering} there exists a morphism $L \to K$ of hypercoverings of $X$ such that $L_{p - q} \to K_{p - q}$ factors through $u$. Then clearly the image of $\xi_q$ in $\underline{H}^q(\mathcal{F})(L_{p - q})$. is zero. Since the spectral sequence of Lemma \ref{lemma-cech-spectral-sequence} is functorial this means that after replacing $K$ by $L$ we reach the situation where $\xi_q, \ldots, \xi_{p - 1}$ are all zero. Continuing like this we end up with a hypercovering where they are all zero and hence $\xi$ is in the image of the map $\check{H}^p(X, \mathcal{F}) \to H^p(X, \mathcal{F})$. \medskip\noindent Suppose that $K$ is a hypercovering of $X$, that $\xi \in \check{H}^p(K, \mathcal{F})$ and that the image of $\xi$ under the map $\check{H}^p(X, \mathcal{F}) \to H^p(X, \mathcal{F})$ of Lemma \ref{lemma-cech-spectral-sequence} is zero. To finish the proof of the theorem we have to show that there exists a morphism of hypercoverings $L \to K$ such that $\xi$ restricts to zero in $\check{H}^p(L, \mathcal{F})$. By the spectral sequence of Lemma \ref{lemma-cech-spectral-sequence} the vanishing of the image of $\xi$ in $H^p(X, \mathcal{F})$ means that there exist elements $\xi_1, \ldots, \xi_{p - 2}$ with $\xi_q \in \check{H}^{p - 1 - q}(K, \underline{H}^q(\mathcal{F}))$ (more precisely the images of these in certain subquotients) such that the images $d_{q + 1}^{p - 1 - q, q}\xi_q$ (in the spectral sequence) add up to $\xi$. Hence by exactly the same mechanism as above we can find a morphism of hypercoverings $L \to K$ such that the restrictions of the elements $\xi_q$, $q = 1, \ldots, p - 2$ in $\check{H}^{p - 1 - q}(L, \underline{H}^q(\mathcal{F}))$ are zero. Then it follows that $\xi$ is zero since the morphism $L \to K$ induces a morphism of spectral sequences according to Lemma \ref{lemma-cech-spectral-sequence}.