We have the following Setting
I am not sure about how to derive the asympotic distribution of $\sqrt{n} \left(\hat{\beta_{IV}} - \hat{\beta_{OLS}} \right)$. The expression $\sqrt{n} \left(\hat{\beta_{IV}} - \hat{\beta_{OLS}} \right)$ is given by $$\left(\frac{1}{n} \sum_{i=1}^N z_i^{\prime} x_i\right)^{-1} \frac{1}{\sqrt{n}} \sum_{i=1}^N z_i^{\prime} u_i-\left(\frac{1}{n} \sum_{i=1}^N x_i^{\prime} x_i\right)^{-1} \frac{1}{\sqrt{n}} \sum_{i=1}^N x_i^{\prime} u_{i}$$ Then the left part converges to a normal distribution and the right side as well, using central limit theorem, weak law of large numbers and continuous mapping theorem, and slutsky theorem. How do I proceed from here? Thanks in advance!