Proof metric space with distance function

Question

Thats the first time i have to do such an proof but don't know how, never seen or done this before. Especially (iii).

Let $X$ be the Set of all complex sequences. $$ d((a_n),(b_n)) := \sum^\infty_{i=0} \frac{1}{2^{i+1}}\frac{\left | a_i-b_i \right |}{1+\left | a_i-b_i \right|}, ((a_n),(b_n) \in X) $$ Proof that $(X,d)$ is an metric space.

Definition of metric space says:

1. $d((a_n),(b_n)) \geq 0 $ and $d((a_n),(b_n))=0 \Leftrightarrow (a_n)=(b_n)$
2. $d((a_n),(b_n)) = d((b_n),(a_n))$
3. $d((a_n),(c_n) \leq d((a_n),(b_n))+d((b_n),(c_n)) \ Triangle \ inequality $

Can someone help me please

Thanks
Landau.