Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 113 / 4 / a

If the germ of a sheaf section $s_p$ is zero, then by the definition of a stalk of a sheaf there is a neighbourhood $V$ of $p$ on which $s{|}_V=0$. Every point of $V$ also has zero germ. Thus the complement of $\mathrm{Supp}s$ is open, so the support of a sheaf section is closed.