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

Suppose $f$ satisfies that for each $\varepsilon >0$,there exists $\delta>0$ such that for all $x,y\in X$, $$\varepsilon\leq d(x,y)<\varepsilon+\delta\implies d(f(x),f(y))<\varepsilon.$$

Is $\{x_n\}$ a Cauchy sequence ?