I am interested in how the estimate of the regression $\beta_1$ decreases with respect to sample size $n$. Does this look ok or am I missing something? $$ \begin{align} \text{Var}(\hat \beta_1) & = \frac{\sigma^2}{\sum_{i=1}^n (x_i - \bar x)^2} \\ & \le \frac{\sigma^2}{\sum_{i=1}^n \min_i \{(x_i - \bar x)^2\}} \\ & = \frac{\sigma^2}{n \min_i\{(x_i - \bar x)^2\}} \\ & = O\bigg(\frac{1}{n}\bigg). \end{align} $$
Edit: It seems that the first answer in this post seems to agree with my result. However, the first answer in this post seems to say something else as it says that $\sigma^2$ is itself estimated (by the MSE) and that this estimate has a dependence on $n$.
Can someone clarify exactly how the variance decreases with $n$?