How do you prove Cov $\left( \bar{X} , X\_i - \bar{X} \right) = 0$ given $ X\_1 ,..., X\_i$ are i.i.d. each with variance $\sigma^2$ and $\bar{X}$ is the sample mean?? In other words, how do you show that the sample mean and the differences of the observations from that mean are not linearly correlated???

Am I on the write track? 
I've written

$Var \left( \bar{X} + \left( X\_i - \bar{X} \right) \right) = Var X\_i = Var \bar{X} + Var \left( X\_i - \bar{X} \right) + Cov \left( \bar{X} , X\_i - \bar{X} \right)$ . 

I know 
$Var X\_i = \sigma^2 , Var \bar{X}= \frac{\sigma^2}{n}$, so all I need left to find the covariance is $Var \left( X\_i - \bar{X} \right)$. 

I don't know if this is right, but I wrote

$Var \left( X\_i - \bar{X} \right) = E\left( \left( \left( X\_i - \bar{X} \right) - \mu\_{ X\_i - \bar{X}} \right)^2 \right)=E \left( \left( X\_i - \bar{X} \right)^2 \right)=s^2$, but I don't know if that's right or helpful at all. 

Am I on the right track??