so I need to be able to prove that, given the residual is given by ei=yi-y(hat)i, the mean of the residuals, ie e-bar, is always equal to zero
but that's false, if your model is $\displaystyle Y=\beta X+\epsilon$
However it is true if the model does include the constant term, say $\displaystyle Y=\beta_0+\beta_1 X+\epsilon$
In that second case, which using matrices is a lot easier we have... from http://en.wikipedia.org/wiki/Regression_analysis
$\displaystyle \sum(y_i-\hat y_i)=\sum(y_i-\hat \beta_0-\hat\beta_1 x_i)$
$\displaystyle =\sum(y_i-\bar y+\hat \beta_1\bar x-\hat\beta_1 x_i)$
$\displaystyle =\sum(y_i-\bar y)-\hat \beta_1\sum(x_i-\bar x)$
$\displaystyle =0-0=0$