Let $\vec{X}=(X_1,X_2,\ldots,X_n)$ be a random sample of some random variable $X$ whose distribution $F$ depends on some real-valued parameter $\theta_F$. Let $\hat\theta(\vec{X})$ be an unbiased estimator of $\theta_F$. Let $(\vec{x}_n)$ be a sequence of observations of $\vec{X}$. Is it true that the sequence $(\hat\theta(\vec{x}_n))$ will have $\theta_F$ as a cluster point?
My guess is that it should be true, but I am not sure how to prove this from the fact that the estimator is unbiased. Any tips?