Convergence in Probability

Hello all,

Here is my problem:

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

Show that $\displaystyle \frac{1}{n}\sum_{i=1}^{n} (X_i-\bar{X})^2\rightarrow\sigma^2$ 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 $\displaystyle (X_i-\bar{X})^2$ is an iid random variable with finite variance and expected value $\displaystyle \sigma^2$, so then I could use WLLN to say the sum converges in probability to $\displaystyle \sigma^2$. But when I try that, I find that $\displaystyle (X_i-\bar{X})^2$ is not iid.

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