I want to know if the following statement holds. It is sort of the converse of the CLT. Intuitively, my guess is that it is true. However, I have no idea on how to prove it.
Suppose that:
$$\sqrt{n}(\hat{\theta_n}-\theta)\overset{d}{\to}N(0,\sigma^2) $$
Where $\theta \in \mathbb{R}$. Does it follow that $\hat{\theta_n}\overset{d}{\to}\theta$?
What are you thoughts?