Fix a perfect set $D\subseteq[0,1]$. Does there necessarily exist a monotone sequence $\{x_m\}_{m\in\mathbb{N}}\subseteq D$ such that $x_m\rightarrow x\in D$ with the property that: \begin{equation} \frac{x_m-x_{m-1}}{x_{m-1}-x_{m-2}}\rightarrow c \end{equation} for some $c\in(0,\infty)$?
As all elements of $D$ are accumulation points, this seems natural, but I could not find a reference to such a property nor find a clear counterexample. Any help is much appreciated!
For example, the Cantor set satisfies this property. Take $x_n=(1/3)^n$. For all $n\in\mathbb{N}$, $x_n$ is in the Cantor set. So too is $0$. Moreover, the ratio of the increments is always c=1/3.
