## Convergence in Probability

Hello all,

Here is my problem:

Let $X_n$ be iid, $E(X_n)=\mu, Var(X_n)=\sigma^2$.

Show that $\frac{1}{n}\sum_{i=1}^{n} (X_i-\bar{X})^2\rightarrow\sigma^2$ in probability.

This seems like it should be easy. It's in the section from the book on the Weak Law of Large Numbers (among other things.) I thought I should show that $(X_i-\bar{X})^2$ is an iid random variable with finite variance and expected value $\sigma^2$, so then I could use WLLN to say the sum converges in probability to $\sigma^2$. But when I try that, I find that $(X_i-\bar{X})^2$ is not iid.

I feel like this is easy and I'm just missing something basic. Any thoughts?