Newton's Method/Sequence of Approximations Converges Quadratically

Theorem
Let $\map f x$ be a real function.

Let $\alpha$ be a root of $\map f x$.

Let $\epsilon > 0$ be a positive real number, and $I = \closedint {\alpha - \epsilon} {\alpha + \epsilon}$.

Let $f$ have a continuous second derivative on $I$.

Let $\ds M = \map \sup {\size {\frac {\map {f' '} s} {\map {f'} t} } }$ over all $s, t \in I$.
 * For this to be well-defined, it is also necessary that $\map {f'} x$ is non-vanishing on $I$.

Suppose $\epsilon < \dfrac 2 M$.

Then the sequence generated by Newton's Method, starting with any initial guess $x_0 \in I$, converges to $\alpha$ with order $2$.

Proof
Suppose that the sequence is produced up to $x_n$.

Suppose also that $x_n \in I$.

First:

Therefore $x_{n + 1} \in I$.

But as $x_0 \in I$, by Principle of Mathematical Induction, every $x_i \in I$.

Next, apply the Principle of Recursive Definition to define the sequence:

Trivially:
 * $\size {\alpha - x_0} \le \epsilon_0$

Assuming $\size {\alpha - x_n} \le \epsilon_n$, it follows that:

By Principle of Mathematical Induction, every $\size {\alpha - x_n} \le \epsilon_n$.

But:

Therefore, the sequence $\sequence {x_n}$ converges to $\alpha$ with order $2$, by definition.