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 $\ds M = \map \max {\size {\frac {\map {f' '} s} {\map {f'} t} } }$ 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 that $\map f x$ is $C^2$ on $I$.

That is, the second derivative is continuous on the interval.

Suppose also that $2 \epsilon M < 1$.

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