Prove that the order of convergence must be $\geq 1$ Given a sequence $\{x_n\}_n$ and real numbers $c > 0$ and $L$, such that $\displaystyle\lim_{n \to \infty}x_n - L = 0$ and $\displaystyle\lim_{n \to \infty} \frac{| x_{n+1} - L |}{|x_n - L|^p} = c$, prove that $p \geq 1$.
This is assumed without proof in my textbook and I'd like a rigorous one, but I can't come up with it.

If we allow $c = 0$, then this is not true.
Take $$x_n = \frac{1}{n}$$
where we can pick $p = 0.5$
If $c \gt 0$, then I believe it is true.
We use the following fact:
If $f_n \to \infty$ and if $\dfrac{f_{n+1}}{f_n} \to p$, then $p \ge 1$.
(because otherwise $\sum f_n$ would be absolutely convergent, by the ratio test!)
Assuming $x_n \neq L$ for any $n$ (otherwise problem is meaningless I suppose).
Picking $f_n = -\log |x_n - L|$ will give the result, as I believe we can then show that $\dfrac{f_{n+1}}{f_n} \to p$.
