This is a proof question that I just cannot get what's required for the life of me. This isn't part of my assessed homework, just fyi.

*Suppose $\displaystyle X_1$, $\displaystyle X_2$, ..., $\displaystyle X_n$ is a random sample from a random variable $\displaystyle X$ which has expectation $\displaystyle \mu$ and variance $\displaystyle \sigma^2$. Consider the sample variance* *$\displaystyle S^2=\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2$* *Show that $\displaystyle E[S^2]=\sigma^2$*
We're also given the hint that $\displaystyle X_i-\bar{X}$ may be written as $\displaystyle (X_i-\mu)-(\bar{X}-\mu)$ and that $\displaystyle Var(X)=E[(X-\mu)^2]$. I've used both of these but I just can't seem to get $\displaystyle \sigma^2$ at all. Closest I've gotten is $\displaystyle \frac{n\sigma^2}{n-1}$