Rate of convergence conditions

Question

Given a convergent sequence $(x_k)$ does it suffice that $$|x_{k+1}|\le C|x_k|+o(|x_k|)$$ definitively, with $|C|<1$ to state that the rate of convergence of $(x_k)$ is less than $C$?

Answer 1

If you assume $x_k\to0$, then the rate of convergence can be estimated as follows:

$$ \lim \frac{|x_{k+1}|}{|x_k|} \le \lim \frac{C|x_k|+\mathrm o(|x_k|)}{|x_k|} = C+\underbrace{\lim \frac{\mathrm o(|x_k|)}{|x_k|}}_{\to\,0\,\text{by assumption}}=C $$

So you can conclude that it is less or equal $C$. You cannot do better because $x_n=C^n$ would be a counter-example (here $\mathrm o(|x_k|)=0$).