In the following slide ECON4150 - Introductory Econometrics Lecture 16: Instrumental variables, Monique de Haan
it says that "instrument exogeneity implies $E[u_i \mid Z_i]=0$" where instrument exogeneity is defined on slide 13 as $cov(Z_i, u_i) =0$. Here $Z_i$ is the instrument and $u_i$ is the error term in the structural model with a single endogenous variable.
Can someone prove to me why this is true? I thought the implication only held the other way, i.e., $E[u_i \mid Z_i] = 0$ implies $cov(Z_i, u_i)=0$ but the slides suggests that $cov(Z_i, u_i)=0$ implies $E[u_i \mid Z_i] = 0$, which isn't true in general?