Derive zero conditional mean $E(u|x)=E(ux)=0$ in OLS

Question

Here is the capture from Youtube vedio, I've seen assumptions as $E(u)=0,\,E(u|x)=0$, but I have no idea where does $E(ux)=0$ comes from, how to derive this one.

Appreciate for any help

Zero conditional mean of errors

Answer 1

Assume that $E[u|X]=0$ holds and use the law of iterated expectation $$ E[Xu] = E[E[Xu|X]] = E[X[E[u|X]]=E[X\cdot 0] = 0 $$