In a statistics book i'm reading, it is postulated that asymptotic normality of an estimator implies consistency. That is $$ \hat{\theta}_n \stackrel{as}{\sim} \mathcal{N}(\theta_0, \frac{1}{n}\sigma(\theta_0)) \Rightarrow P_{\theta_0}(|\hat{\theta}_n - \theta_0|>\epsilon ) \to 0 $$ when $n\to\infty$, for all $\theta_0 \in \Theta$ and $\epsilon>0$.
I am trying to prove this, but i can't seem to get a breakthrough.
If anyone could shed some light on how this is proven, i would be very grateful.