Newton's Method: Quadratic Convergence Proof and Stopping Rule

Let $f : \mathbb{R} \to \mathbb{R}$ be twice continuously differentiable, and suppose $f(\alpha) = 0$ with $f'(\alpha) \neq 0$ (i.e., $\alpha$ is a simple root). 1. Write down Newton's method for approximating $\alpha$. 2. Prove that Newton's method converges locally and quadratically: if $x_0$ is sufficiently close to $\alpha$, then $|x_{n+1} - \alpha| \leq C |x_n - \alpha|^2$ for some constant $C$. 3. State a practical stopping rule that controls the absolute error in the root estimate.