Let $C$ be a nonempty subset of a Banach space $X$. A mapping $T:C\to C$ is said to be asymptotically nonexpansive mappings if for all $n\in \mathbf{N},$ there exists a positive constant $k_n\geq1$ such that $\lim\limits_{n\to \infty}k_n=1$ $$\|T^nx-T^ny\|\leq k_n\|x-y\| \ \ \ \text{for all}x, y\in C.$$
Question
I think if $T$ be a nonexpansive mappings ($\|Tx-Ty\|\leq \|x-y\|$) then it will be asymptotically nonexpansive mappings. But I cannot prove that there is an asymptotically nonexpansive mappings, but not nonexpansive.