Mean-independence implies zero covariance

Intuitively: a (population) regression of u on x is the linear approximation to E (u|x). The regression would yield a slope coefficient of Cov (u, x) /V (x). If E (u|x) = E (u), the conditional expectation function is a constant. Therefore, a linear approximation to it would yield a slope of zero, which implies that Cov(u, x) =0.

Formally,
E (u|x) = E(ux|x)/x and therefore E (ux|x) = xE (u|x).

But the law of iterated expectations implies that E (ux) =E [E (ux|x)] . Therefore,
E (ux) = E [E (ux|x)] = E [xE (u|x)]

If E (u|x) = E (u), we have that E (ux) = E [xE (u|x)] = E [xE (u)] = E (x)E (u)

Because Cov (u, x) = E (ux) − E (x)E (u), we have shown that E (u|x) = E (u) implies that
Cov (u, x) = 0.