Let $S$ be the set of all real sequences $x=\{x_n\}$, $d: S\times S \rightarrow \mathbb R$ be defined by:
$$d(x,y)=\sum_{n=1}^{\infty} \frac{|x_n-y_n|}{2^{n}[1+|x_n-y_n|]}.$$
Show that $(S,d)$ is a complete metric space.
I want to say we can use the Cantor Intersection property or an isometry here to prove completeness. However, I am having trouble getting started. Any suggestions?