If $\displaystyle x_n $ is a sequence and there is a $\displaystyle c \geq 1 $ such that $\displaystyle |x_{k+1}-x_{k}| > c|x_{k}-x_{k-1}| $ for all $\displaystyle k > 1 $, then can $\displaystyle x_n $ converge?

We claim that if $\displaystyle n \in \bold{N} $, then $\displaystyle |x_{k}-x_{k+1}| > c^{k-1}|x_{1}-x_{2}| $. For $\displaystyle k = 1 $, $\displaystyle |x_{1}-x_{2}| > |x_{1}-x_{2}| $, which is false. So it cannot be a contractive sequence?