If we know that $\hat\theta_n\overset{p}\to\theta_0$, how does the following equation \begin{equation} \sqrt{n}V_{\theta_0}\cdot(\theta_0-\hat\theta_n)+\sqrt{n}o_p(|\hat\theta_n-\theta_0|)=G_n\psi_{\theta_0}+o_p(1)\qquad\qquad(1) \end{equation} become $$\sqrt{n}V_{\theta_0}\cdot(\hat\theta_n-\theta_0)=-G_n\psi_{\theta_0}+o_p(1)?\qquad\qquad(2)$$
This is part of proof for Theorem 5.21 on page 52-53 in Van der Vaar(1998). I wonder how can the term $\sqrt{n}o_p(|\hat\theta_n-\theta_0|)$ in equation (1) be omitted. The author also provide an inequality between these two equations. That is, $$\sqrt{n}|\hat\theta_n-\theta_0|\le|V_{\theta_0}^{-1}|\sqrt{n}|V_{\theta_0}(\hat\theta_n-\theta_0)|=O_p(1)+o_p(\sqrt{n}|\hat\theta_n-\theta_0|)\qquad\quad (3)$$
He also writes, "This implies that $\hat\theta_n$ is $\sqrt{n}$-consistent: The left side is bounded in probability. Inserting this in equation (1), we obtain equation (2)."