Help me to understand the proof that is given in the "Numerical Optimizaton" by Jorge Nocedal & Stephen J. Wright. The theorem says that at a near point $x_0$ then $\{x_k\}\to x^*$ by taking $x_{k+1}=x_k+p_{k}$ where $p_k$ is a descent direction and $\nabla^2f_kp_k=-\nabla f_k$ and the rate of convergence is quadratic and $\{\nabla f_k\}\to 0$ also converges quadratically.
After the step (3.32), (Since the Hessian is non singular there is a radius...) I get lost, can someone help me unserstanding this?