If $T$ is a complete sufficient statistic, then $Cov(T, U)=0$ for all unbiased $U$

Question

I want to prove the following-

Show that if $T$ is complete sufficient for $θ$, then $Cov_θ(T, U) = 0$ for all $θ ∈ Θ$ and for all $U$ satisfying $E_θ(U) = 0$ for all $θ ∈ Θ$. I think in essence it means that the complete sufficient statistic of any parameter is uncorrelated with all unbiased estimators of that parameter.

I tried to consider the Rao-Blackwellization of $U$ with respect to $T$, and I tried to define a new estimator $ U^* = \mathbb{E}[\mathbb{E}[U|T]]$ and now $E(U^*)=0$, but I am not able to proceed further.

It'll be great if someone can help, thanks!