Heteroscedasticity and weighted least square estimator
Let $\hat{\beta}$ be the OLS estimator of $\beta$ in
$$y_t = \beta x_t + u_t$$
Let $\tilde{\beta}$ be the OLS estimator of $\beta$ in
$$\dfrac{y_t}{\sqrt{k_t}} = \beta \dfrac{x_t}{\sqrt{k_t}} + \dfrac{u_t}{\sqrt{k_t}}$$
$\text{Var}(\hat{\beta}) = \dfrac{\sum x_t^2\sigma_t^2}{\left(\sum x_t^2\right)^{2}} $

$\text{Var}(\tilde{\beta}) = \dfrac{\sigma^2_u}{\sum \dfrac{x_t^2}{k_t}} $

So, $\dfrac{\text{Var}(\tilde{\beta})}{\text{Var}(\hat{\beta})} = \dfrac{\left(\sum x_t^2\right)^{2}}{\sum \dfrac{x_t^2}{k_t} \sum {x_t^2}{k_t}} $.

Let $a_t = \dfrac{x_t}{\sqrt{k_t}}$ and $b_t = {x_t}{\sqrt{k_t}}$, and we can rewrite the ratio of variances as

$\dfrac{\text{Var}(\tilde{\beta})}{\text{Var}(\hat{\beta})} = \dfrac{\left(\sum a_tb_t\right)^{2}}{\sum a_t^2 \sum {b_t^2}} \leq 1 $ [By Cauchy-Schwarz inequality]