The Stacks project

Proof. We have to show that given any sheaf $\mathcal{F}$ on $\mathcal{C}$, any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and any distinct sections $s, s' \in \mathcal{F}(U)$, there exists a point $p$ such that $s, s'$ have distinct image in $\mathcal{F}_ p$. See Lemma 7.38.3. Consider the system $(J, \geq , V_ j, g_{jj'})$ with $J = \{ 1\}$, $V_1 = U$, $g_{11} = \text{id}_ U$. Apply Lemma 7.39.2. By the result of that lemma we get a system $(I, \geq , U_ i, f_{ii'})$ refining our system such that $s_ p \not= s'_ p$ and such that moreover for every finite covering $\{ W_ k \to W\}$ of the site $\mathcal{C}$ the map $\coprod _ k u(W_ k) \to u(W)$ is surjective. Since every covering of $\mathcal{C}$ can be refined by a finite covering we conclude that $\coprod _ k u(W_ k) \to u(W)$ is surjective for any covering $\{ W_ k \to W\}$ of the site $\mathcal{C}$. This implies that $u = p$ is a point, see Proposition 7.33.3 (and the discussion at the beginning of this section which guarantees that $u$ commutes with finite limits). $\square$