This is an exercise in Vakil's Foundations of Algebraic Geometry, namely 5.2.A.
Let $X$ be a reduced scheme. If $a\in \mathscr{O}_X(X)$ is such that its image in $\mathscr{O}_{X,p}$ lies in the maximal ideal $\mathfrak{m}_{p}$ for each $p\in X$, then $a = 0$.
I am able to show that a scheme is reduced iff $\mathscr{O}_{X,p}$ is reduced for each $p\in X$, but I am not sure how to use this to prove the above statement.