I have this model without intercept
$y_i = \beta x_i + u_i$
where $\widehat{\beta} = \frac{\sum x_i y_i}{\sum x_i^2} = \beta + \frac{\sum x_i u_i}{\sum x_i^2}$
I want to verify the consistency of $\widehat{\beta}_1$.
¿Is it correct to say that $\sum x_i u_i \rightarrow Cov(x,u)$ and $\sum x_i^2 \rightarrow V(x)$?
Can you explain me why, please. Shouldn't they be in deviations from the mean?