Let $S$ denote the set of all sequences of real numbers. For $x=(x_1,x_2,x_3,...)$, $y=(y_1,y_2,y_3,...)$ in $S$, consider $$d(x,y)=\sum_{i=1}^\infty \frac{1}{2^i}\frac{|x_i-y_i|}{1+|x_i-y_i|}$$ I need to show that $d$ is a metric on the space $S$.
I have verified the 3 properties of the metric, but how would I show that $d(x,y) \le d(x,z) +d(z,y)?$
I was given a hint and to try show that for $a,b,c \in \mathbb R$
$$\frac{|a-b|}{1+|a-b|}\le \frac{|a-c|}{1+|a-c|} + \frac{|c-b|}{1+|c-b|} (1)$$
I tried multiplying $(1)$ by $(1+|a-b|)(1+|a-c|)(1+|c-b|)$ to get rid of the denominators, but either way I cant seem to get anywhere. How would i go about proving that $d(x,y) \le d(x,z) +d(z,y)?$