I am given a situation where $X_1,X_2,...X_n \sim_{iid} X$ with pdf $f_X(x;\theta)$ and $u(X)$ is an unbiased and sufficient estimator of $\theta$.
I am working on proofs where I am trying to find if $X$ is complete and it seems that as long as
$E[u(X)]=0$ is given and you can show that $E[X^ku(X)]=0$ for any natural number $k$ then you can show $u(X)=0$.
I do not know why the last logic works.
May I have some help please?