# How to prove that $\hat\theta_n$ is a consistent estimator of $\theta$ if and only if $a_n \rightarrow \infty$ as $n \rightarrow \infty$

Suppose that $$\hat\theta_n, n \in \mathbb{N}$$, is a sequene of estimators of $$\theta \in \mathbb{R}$$ such that $$a_n (\hat\theta_n - \theta) \xrightarrow{d} \mathcal{N}(0, \sigma^2)$$ for some sequence of positive real numbers $$a_n, n \in \mathbb{N}$$, and $$\sigma^2 > 0$$. How would you prove that $$\hat\theta_n$$ is a consistent estimator of $$\theta$$ if and only if $$a_n \rightarrow \infty$$ as $$n \rightarrow \infty$$.

Hint: Use, $$\hat{\theta}_n - \theta = \dfrac{1}{a_n}a_n(\hat{\theta}_n - \theta)$$
Consider $$\dfrac{1}{a_n}$$ and $$a_n(\hat{\theta}_n - \theta)$$ separately since you know the asymptotic behavior of each quantity.