Let $(X,d)$ be a complete metric space and $f$ a mapping of $X$ into itself.$\{f^n(x)\}=\{x_n\}$ sequence of iterated transforms.

And $f$ satisfies :for each $\varepsilon >0$,there exists $\delta>0$ such that for all $x,y\in X$, $$\varepsilon\leq d(x,y)<\varepsilon+\delta\implies d(f(x),f(y))<\varepsilon.$$ Does $\{x_n\}$ a Cauchy sequnce ?