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.