$X_1, X_2, ... , X_n$ are iid $N(0, \theta)$ random variables with $\theta$ in $(0, \infty)$. With the prior distribution $\pi(\theta)$=$\frac{4e^\frac{-2}{\theta}}{\theta^3}$, I calculated the posterior distribution and used the squared error loss function to obtain the Bayes estimator for $\theta$ (the mean of the posterior distribution). I found the Bayes estimator to be $\frac{4 + \sum x_i^2}{n+2}$. How do I show that this estimator is consistent?
I know showing the estimator is consistent has something to do with showing convergence in probability, however, I am completely lost on where to start.