Let $Y_{i}=\beta_{1}+\beta_{2}X_{i}+u_{i}$ be the underlying linear relationship between observed values $X_{i}$ and $Y_{i}$ with $u_{i}$ being randomness and/or disturbance term.
Let $\hat\beta_{2}$ be an OLS estimator of the underlying parameter $\beta_{2}$.
When computing $E(\hat\beta_{2} u_{i})$ and $Cov(\hat\beta_{2},u_{i})$, I find that the two have equal values such that $E(\hat\beta_{2} u_{i})=Cov(\hat\beta_{2},u_{i})=\frac{\sigma^{2}x_{i}}{\sum x_{i}^{2}}$.
How can I explain this relationship?
Edit (reconsidering my explanation): Both expressions rely on the subscript of $u_{i}$ because $\hat\beta_{2}$'s randomness comes from the presence of $u_{i}$ in its expression. Not sure how to explain why the 2 expressions are equal though.