Denote $X$, the space of all sequences $\in$ $\mathbb R$. I have a metric $d(x,y):=\sum_{n=1}^\infty 2^{-n}\frac{| x_n-y_n|}{1+| x_n-y_n|}$
If $(X,d)$ is a metric space and if $A$ is a subset of $X$, the diameter of $A$ is defined as $\operatorname {diam} \left({A}\right) := \sup \left\{{d \left({x, y}\right): x, y \in A}\right\}$. I know that the set $A$ is bounded if it has finite diameter. What Im stuck at is how would I show that $\operatorname {diam} \left({A}\right) \le 1$?