Our teacher proved in class that if the best unbiased estimator exists, then it is an MLE using a theorem that if $\hat{\theta}-\theta$ is proportional to the score of $\theta$ with probability $1$, then $\hat{\theta}$ is the best unbiased estimator since it attains the CR bound.
In general, I know that MLE attains the CR bound asymptotically. So I'm in doubt whether the statement in the title holds for a finite sample. Could anyone provide some insight about relationship between the best unbiased estimator and MLE in finite sample case (and proof)?