a) Show that $E\{X-E(X)\} = 0$ for any random variable $X$.
b) Use the result in part (a) and the following equation to show that if two random variables are independent then they are uncorrelated,
If $X$ and $Y$ are independent, then for all functions $g$ and $h$, $E\{g(X)h(Y)\} = E\{g(X)\}E\{h(Y)\}$
So I understand the intuition behind a); since $E(X)$ would be the mean, the expectation of any given $X$ minus the mean would be $0$.. but putting that into the form of a proof is eluding me..
Any help or guidance is appreciated :)