The Stacks project

Exercise 111.36.3. Let $X$ be a scheme. Let $x, x' \in X$. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules. Suppose that (a) $x$ is a specialization of $x'$ and (b) $\mathcal{F}_{x'} \not= 0$. Show that $\mathcal{F}_ x \not= 0$.