Can someone could tell me if this is true and give me a proof:
Assume that $(x_k)_k$ converges to a limit $l$.
$ \lim\limits_{x \to 0} \frac{|x_{k+1} - l |}{|x_{k} - l |}= \inf\left\{\mu \in \left[0,1\right] : |x_{k+1} - l | \leq \mu|x_{k} - l | \space \text{for} \space k \space \text{large enough}\right\} $
I failed to prove it, using the definition of the infimum. (it is an alternative definition of convergence rate).
Thank you!