Why Const? - linear regression - variance

Question

the "picture" is part of the demonstration of the formula of Variance of the slope coefficient in Linear Regression, my question is why the first part in circle is equal to Const?

Answer 1

Note that \begin{align} \hat{\beta}_0 &= \beta_0 \sum \frac{1}{n} + \beta_1 \bar{x} + \beta_0 \bar{x} \sum w_i - \beta_1 \bar{x} \sum x_i w_i + \sum (x_i - \bar{x}w_i)\epsilon_i\\ &= \beta_0 + \beta_1 \bar{x} + 0 - \beta_1\bar{x} + \sum (x_i - \bar{x}w_i)\epsilon_i\\ &=\beta_0 + \sum (x_i - \bar{x}w_i)\epsilon_i\\ & = \beta_0 + \sum v_i\epsilon_i\\ & = \beta_0 + o_p(1). \end{align} When $\beta_0$ is parameter and the "error term" converges to $0$ in probability.