Consider the simple linear regression model $y=\beta_0+\beta_1x+\epsilon$ with $\epsilon \sim\ \text{NID}(0,\sigma^2)$.
Show that the least squares estimators $\hat{\beta_0}$ and $\hat{\beta_1}$ are uncorrelated.
My attempt:
Since $\hat{\beta_0}=\bar{y}-\hat{\beta_1\bar{x}}$ and $\hat{\beta_1}=\dfrac{S_{xy}}{S_{xx}}=\displaystyle\sum\limits_{i=1}^n\dfrac{x_i-\bar{x}}{S_{xx}}$. Then $$\begin{aligned} \text{Cov}(\hat{\beta_0}, \hat{\beta_1})&=\text{Cov}(\bar{y}-\hat{\beta_1\bar{x}}, \hat{\beta_1})\\ &=\text{Cov}(\bar{y}, \hat{\beta_1})-\text{Cov}(\hat{\beta_1\bar{x}},\hat{\beta_1})\\ &=\text{Cov}(\bar{y}, \hat{\beta_1})-\bar{x}\cdot\text{Cov}(\hat{\beta_1\bar{x}},\hat{\beta_1})\\ \end{aligned}$$ Now, $$\begin{aligned} \text{Cov}(\bar{y}, \hat{\beta_1})&=\text{Cov}\left(\dfrac{1}{n}\displaystyle\sum\limits_{i=1}^ny_i,\sum\limits_{i=1}^n\frac{x_i-\bar{x}}{S_{xx}}y_i\right)\\ &=\frac{1}{n}\sum\limits_{i=1}^n\frac{x_i-\bar{x}}{S_{xx}}\cdot\text{Var}(y_i)\\ &=\frac{\sigma^2}{n}\sum\limits_{i=1}^n\frac{x_i-\bar{x}}{S_{xx}}\qquad \text{,Var($y_i$)$=\sigma^2$}\\ &=0\qquad, \text{Since $\sum\limits_{i=1}^n\frac{x_i-\bar{x}}{S_{xx}}=0$} \end{aligned}$$
Also, $$\begin{aligned} \bar{x}\text{Cov}(\hat{\beta_1},\hat{\beta_1})&=\bar{x}\text{Var}(\hat{\beta_1})\\ &=\bar{x}\cdot\frac{\sigma^2}{S_{xx}}\qquad, \text{Since $\text{Var}(\hat{\beta_1})=\frac{\sigma^2}{S_{xx}}$} \end{aligned}$$ Thus, $$\begin{aligned} \text{Cov}(\hat{\beta_0}, \hat{\beta_1})&=0-\frac{\bar{x}\sigma^2}{S_{xx}} \end{aligned}$$ How to conclude that $\hat{\beta_0}$ and $\hat{\beta_1}$ are uncorrelated? if they are not equal to $0$.