I am trying to work out the derivation of the variance of the sample mean. It is clear to me how the normal proof using $\frac{1}{n^2}(n\sigma^2)$ works. However, I was trying to work out how to do the same proof using the good old $Var(X) = E[(X-E[X])^2]$ and couldn't quite work it out. There must be something fundamental I am missing. What I have worked out so far is this:
$$ Var(\overline{x}) = E\left[\left(\frac{\sum_i x_i}{n}-E\left[\frac{\sum_i x_i}{n}\right]\right)^2\right]\\ = E\left[\left(\frac{\sum_i x_i}{n}\right)^2-2*\frac{\sum_i x_i}{n}*E\left[\frac{\sum_i x_i}{n}\right]+E\left[\frac{\sum_i x_i}{n}\right]^2\right] \\ = E\left[\left(\frac{\sum_i x_i}{n}\right)^2\right]-E\left[\frac{\sum_i x_i}{n}\right]^2 $$
I think the latter term can be re-written as $\frac{1}{n}E[x]^2$, but I am struggling with the rest. How do I do this?