I'm doing problem 22-14 in Spivak's Calculus, 4th edition. Here they outline Newton's method. They assume for convenience that $f'>0$ and $f''>0$, and that $f(x_1)>0$. They note that in this case, if $c< x_1$ is a zero, we have $x_1>x_2>x_3 > \dotsb > c$. Then some analysis follows, and they show that if $$x_1-c < \frac{\inf\limits_{x\in [c,x_1]}f'(x)}{\sup\limits_{x\in [c,x_1]}f''(x)}, \tag{*}$$
then Newton's method will work.
Why is the assumption (*) necessary? Already we have $x_n-c$ a positive decreasing sequence, and so it must reach a limit. Plugging into $$x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)}$$ shows that such a limit must be a zero. Am I missing something, or is this assumption extraneous?